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For the simplest possible example:

t = 1 second

T^1 = 100 cs

T^2 = (1 × 100) / (1 / 100) = 100² = 10000 cs²

A = 1 × 100 = 100 centiseconds (cs)

B = 1 / 100 = 0.01 hectoseconds (hs)

Note how 100, 1 and 0.01 all equal the same length of time.

Equipment.

You need a calculator that converts decimals to fractions to practice mathematics and time. I recommend the Natural Scientific Calculator.

https://8uvp.app.link/FeZPwAXF1V

Requirements.

There are just a few very simple things you need to know to practice maths and time. You can learn them all in seconds.

1. Orders of magnitude (time).

https://mathsandtime.com/orders-of-magnitude-time/

As in attosecond (as) and exasecond (Es) etc. The above link is used for quick reference.

2. Scientific notation.

https://www.purplemath.com/modules/exponent3.htm

As in 2.3855886103 × 10^13

X = 2.3855886103

3. Basics rules of exponents.

https://www.purplemath.com/modules/exponent.htm

As in when you multiply two exponents you add the exponents.

X × 10^13 × 10^6 = X × 10^19

And when you divide two exponents you subtract the exponents.

X × 10^6 / 10^13 = X × 10^-7

That is about it!

Question.

An example of a square of time:

(ORDERS OF MAGNITUDE AND EXPONENTS)

B = A/T^2 = X × 10^4 Ms

What are the magnitudes of T^1, T^-1, T^2 and T^-2?

What are the powers n of X × 10^n for A and t?

Answer:

T^1 = √(A/B) = 10^6 μs

T^-1 = √(B/A) = 10^-6 Ms

T^2 = A/B = 10^12 μs²

T^-2 = B/A = 10^-12 Ms²

A = BT^2 = X × 10^4 × 10^12 = X × 10^16 μs

t = A/T^1 = X × 10^16 / 10^6 = X × 10^10 s

It is an inverse square of time!
For the simplest possible example: t = 1 second T^1 = 100 cs T^2 = (1 × 100) / (1 / 100) = 100² = 10000 cs² A = 1 × 100 = 100 centiseconds (cs) B = 1 / 100 = 0.01 hectoseconds (hs) Note how 100, 1 and 0.01 all equal the same length of time. Equipment. You need a calculator that converts decimals to fractions to practice mathematics and time. I recommend the Natural Scientific Calculator. https://8uvp.app.link/FeZPwAXF1V Requirements. There are just a few very simple things you need to know to practice maths and time. You can learn them all in seconds. 1. Orders of magnitude (time). https://mathsandtime.com/orders-of-magnitude-time/ As in attosecond (as) and exasecond (Es) etc. The above link is used for quick reference. 2. Scientific notation. https://www.purplemath.com/modules/exponent3.htm As in 2.3855886103 × 10^13 X = 2.3855886103 3. Basics rules of exponents. https://www.purplemath.com/modules/exponent.htm As in when you multiply two exponents you add the exponents. X × 10^13 × 10^6 = X × 10^19 And when you divide two exponents you subtract the exponents. X × 10^6 / 10^13 = X × 10^-7 That is about it! Question. An example of a square of time: (ORDERS OF MAGNITUDE AND EXPONENTS) B = A/T^2 = X × 10^4 Ms What are the magnitudes of T^1, T^-1, T^2 and T^-2? What are the powers n of X × 10^n for A and t? Answer: T^1 = √(A/B) = 10^6 μs T^-1 = √(B/A) = 10^-6 Ms T^2 = A/B = 10^12 μs² T^-2 = B/A = 10^-12 Ms² A = BT^2 = X × 10^4 × 10^12 = X × 10^16 μs t = A/T^1 = X × 10^16 / 10^6 = X × 10^10 s It is an inverse square of time!
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  • QUESTION 25.

    TRIGONOMETRIC EQUATIONS.

    θ = s/60 × 2π

    8sin^2(θ) + 2sin(θ) - 1 = 0

    2cos^2(θ) + √3cos(θ) = 0

    a. Solve for θ and s. (Note: there are several possible θ values, find the specific angle that corresponds to both sine and cosine).

    8sin^2(θ) + 4sin(θ) - 2sin(θ) - 1 = 0

    4sin(θ)(2sin(θ) + 1) - 1(2sin(θ) + 1) = 0

    (2sin(θ) + 1) (4sin(θ) - 1) = 0

    2sin(θ) + 1 = 0

    2sin(θ) = -1

    sin(θ) = -1/2

    4sin(θ) - 1 = 0

    4sin(θ) = 1

    sin(θ) = 1/4

    cos(θ)(2cos(θ) + √3) = 0

    cos(θ) = 0 or 2cos(θ) + √3 = 0

    cos(θ) = 0 or cos(θ) = -√3/2

    sin^-1(-1/2) = π - θ = π/6

    θ = π - sin^-1(-1/2) = 7π/6

    cos^-1(-√3/2) = 2π - θ = 5π/6

    θ = 2π - cos^-1(-√3/2) = 7π/6

    7π/6 = s/60 × 2π

    7/12 = s/60

    s = 7/12 × 60 = 35

    b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 35/60 = 7/12

    k = m - s/60 = 39

    m = k + s/60 = 475/12

    m/60 = 95/144

    k = h - m/60 = 19

    h = k + m/60 = 2831/144

    h/24 = 2831/3456

    k = d - h/24 = 23

    d = k + h/24 = 82319/3456

    d/30 = 82319/103680

    k = M - d/30 = 7

    M = k + d/30 = 808079/103680

    M/(73/6) = 808079/1261440

    k = y - M/(73/6) = 50350

    y = k + M/(73/6) = 63514312079/1261440

    c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B.

    T^1 = √(A/B) = 10^18 as

    T^-1 = √(B/A) = 10^-18 Es

    T^2 = A/B = 10^36 as²

    T^-2 = B/A = 10^-36 Es²

    t = A/T^1 = y × 31536000 = 1.587857801975 × 10^12 s

    A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as

    B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es

    d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending.

    M/(73/6) = y - k = 808079/1261440

    M = (y - k) × (73/6) = 808079/103680

    k = M - d/30 = 7

    d/30 = M - k = 82319/103680

    d = (M - k) × 30 = 82319/3456

    k = d - h/24 = 23

    h/24 = d - k = 2831/3456

    h = (d - k) × 24 = 2831/144

    k = h - m/60 = 19

    m/60 = h - k = 95/144

    m = (h - k) × 60 = 475/12

    k = m - s/60 = 39

    s/60 = m - k = 7/12

    s = (m - k) × 60 = 35

    e. Check months with days.

    y - d/365 = t / 31536000 - d/365 = 50350

    d - h/24 = (y - k) × 365 - h/24 = 233

    h - m/60 = (d - k) × 24 - m/60 = 19

    m - s/60 = (h - k) × 60 - s/60 = 39

    s - cs/100 = (m - k) × 60 - cs/100 = 35

    50350 years 233 days 19:39:35

    50350 years 7 months 23 days 19:39:35
    QUESTION 25. TRIGONOMETRIC EQUATIONS. θ = s/60 × 2π 8sin^2(θ) + 2sin(θ) - 1 = 0 2cos^2(θ) + √3cos(θ) = 0 a. Solve for θ and s. (Note: there are several possible θ values, find the specific angle that corresponds to both sine and cosine). 8sin^2(θ) + 4sin(θ) - 2sin(θ) - 1 = 0 4sin(θ)(2sin(θ) + 1) - 1(2sin(θ) + 1) = 0 (2sin(θ) + 1) (4sin(θ) - 1) = 0 2sin(θ) + 1 = 0 2sin(θ) = -1 sin(θ) = -1/2 4sin(θ) - 1 = 0 4sin(θ) = 1 sin(θ) = 1/4 cos(θ)(2cos(θ) + √3) = 0 cos(θ) = 0 or 2cos(θ) + √3 = 0 cos(θ) = 0 or cos(θ) = -√3/2 sin^-1(-1/2) = π - θ = π/6 θ = π - sin^-1(-1/2) = 7π/6 cos^-1(-√3/2) = 2π - θ = 5π/6 θ = 2π - cos^-1(-√3/2) = 7π/6 7π/6 = s/60 × 2π 7/12 = s/60 s = 7/12 × 60 = 35 b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 35/60 = 7/12 k = m - s/60 = 39 m = k + s/60 = 475/12 m/60 = 95/144 k = h - m/60 = 19 h = k + m/60 = 2831/144 h/24 = 2831/3456 k = d - h/24 = 23 d = k + h/24 = 82319/3456 d/30 = 82319/103680 k = M - d/30 = 7 M = k + d/30 = 808079/103680 M/(73/6) = 808079/1261440 k = y - M/(73/6) = 50350 y = k + M/(73/6) = 63514312079/1261440 c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B. T^1 = √(A/B) = 10^18 as T^-1 = √(B/A) = 10^-18 Es T^2 = A/B = 10^36 as² T^-2 = B/A = 10^-36 Es² t = A/T^1 = y × 31536000 = 1.587857801975 × 10^12 s A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending. M/(73/6) = y - k = 808079/1261440 M = (y - k) × (73/6) = 808079/103680 k = M - d/30 = 7 d/30 = M - k = 82319/103680 d = (M - k) × 30 = 82319/3456 k = d - h/24 = 23 h/24 = d - k = 2831/3456 h = (d - k) × 24 = 2831/144 k = h - m/60 = 19 m/60 = h - k = 95/144 m = (h - k) × 60 = 475/12 k = m - s/60 = 39 s/60 = m - k = 7/12 s = (m - k) × 60 = 35 e. Check months with days. y - d/365 = t / 31536000 - d/365 = 50350 d - h/24 = (y - k) × 365 - h/24 = 233 h - m/60 = (d - k) × 24 - m/60 = 19 m - s/60 = (h - k) × 60 - s/60 = 39 s - cs/100 = (m - k) × 60 - cs/100 = 35 50350 years 233 days 19:39:35 50350 years 7 months 23 days 19:39:35
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  • QUESTION 24.

    TRIGONOMETRIC EQUATIONS.

    θ = s/60 × 2π

    3sin^3(θ) + 2 = 13/8

    2cos^2(θ) - √3cos(θ) = 0

    a. Solve for θ and s.

    sin^3(θ) + 2/3 = 13/24

    sin^3(θ) = 13/24 - 2/3 = -1/8

    sin(θ) = ³√(-1/8) = -1/2

    sin^-1(-1/2) = θ - 2π = -π/6

    θ = 2π + sin^-1(-1/2) = 11π/6

    cos(θ)(2cos(θ) - √3) = 0

    cos(θ) = 0 or 2cos(θ) - √3 = 0

    cos(θ) = 0 or cos(θ) = √3/2

    cos^-1(√3/2) = 2π - θ = π/6

    θ = 2π - cos^-1(√3/2) = 11π/6

    11π/6 = s/60 × 2π

    11/12 = s/60

    s = 11/12 × 60 = 55

    b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 55/60 = 11/12

    k = m - s/60 = 47

    m = k + s/60 = 575/12

    m/60 = 115/144

    k = h - m/60 = 2

    h = k + m/60 = 403/144

    h/24 = 403/3456

    k = d - h/24 = 10

    d = k + h/24 = 34963/3456

    d/30 = 34963/103680

    k = M - d/30 = 6

    M = k + d/30 = 657043/103680

    M/(73/6) = 657043/1261440

    k = y - M/(73/6) = 734021

    y = k + M/(73/6) = 925924107283/1261440

    c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B.

    T^1 = √(A/B) = 10^15 fs

    T^-1 = √(B/A) = 10^-15 Ps

    T^2 = A/B = 10^30 fs²

    T^-2 = B/A = 10^-30 Ps²

    t = A/T^1 = y × 31536000 = 2.3148102682075 × 10^13 s

    A = tT^1 = X × 10^13 × 10^15 = X × 10^28 fs

    B = A/T^2 = X × 10^28 / 10^30 = X × 10^-2 Ps

    d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending.

    M/(73/6) = y - k = 657043/1261440

    M = (y - k) × (73/6) = 657043/103680

    k = M - d/30 = 6

    d/30 = M - k = 34963/103680

    d = (M - k) × 30 = 34963/3456

    k = d - h/24 = 10

    h/24 = d - k = 403/3456

    h = (d - k) × 24 = 403/144

    k = h - m/60 = 2

    m/60 = h - k = 115/144

    m = (h - k) × 60 = 575/12

    k = m - s/60 = 47

    s/60 = m - k = 11/12

    s = (m - k) × 60 = 55

    e. Check months with days.

    y - d/365 = t / 31536000 - d/365 = 734021

    d - h/24 = (y - k) × 365 - h/24 = 190

    h - m/60 = (d - k) × 24 - m/60 = 2

    m - s/60 = (h - k) × 60 - s/60 = 47

    s - cs/100 = (m - k) × 60 - cs/100 = 55

    734021 years 190 days 02:47:55

    734021 years 6 months 10 days 02:47:55
    QUESTION 24. TRIGONOMETRIC EQUATIONS. θ = s/60 × 2π 3sin^3(θ) + 2 = 13/8 2cos^2(θ) - √3cos(θ) = 0 a. Solve for θ and s. sin^3(θ) + 2/3 = 13/24 sin^3(θ) = 13/24 - 2/3 = -1/8 sin(θ) = ³√(-1/8) = -1/2 sin^-1(-1/2) = θ - 2π = -π/6 θ = 2π + sin^-1(-1/2) = 11π/6 cos(θ)(2cos(θ) - √3) = 0 cos(θ) = 0 or 2cos(θ) - √3 = 0 cos(θ) = 0 or cos(θ) = √3/2 cos^-1(√3/2) = 2π - θ = π/6 θ = 2π - cos^-1(√3/2) = 11π/6 11π/6 = s/60 × 2π 11/12 = s/60 s = 11/12 × 60 = 55 b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 55/60 = 11/12 k = m - s/60 = 47 m = k + s/60 = 575/12 m/60 = 115/144 k = h - m/60 = 2 h = k + m/60 = 403/144 h/24 = 403/3456 k = d - h/24 = 10 d = k + h/24 = 34963/3456 d/30 = 34963/103680 k = M - d/30 = 6 M = k + d/30 = 657043/103680 M/(73/6) = 657043/1261440 k = y - M/(73/6) = 734021 y = k + M/(73/6) = 925924107283/1261440 c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B. T^1 = √(A/B) = 10^15 fs T^-1 = √(B/A) = 10^-15 Ps T^2 = A/B = 10^30 fs² T^-2 = B/A = 10^-30 Ps² t = A/T^1 = y × 31536000 = 2.3148102682075 × 10^13 s A = tT^1 = X × 10^13 × 10^15 = X × 10^28 fs B = A/T^2 = X × 10^28 / 10^30 = X × 10^-2 Ps d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending. M/(73/6) = y - k = 657043/1261440 M = (y - k) × (73/6) = 657043/103680 k = M - d/30 = 6 d/30 = M - k = 34963/103680 d = (M - k) × 30 = 34963/3456 k = d - h/24 = 10 h/24 = d - k = 403/3456 h = (d - k) × 24 = 403/144 k = h - m/60 = 2 m/60 = h - k = 115/144 m = (h - k) × 60 = 575/12 k = m - s/60 = 47 s/60 = m - k = 11/12 s = (m - k) × 60 = 55 e. Check months with days. y - d/365 = t / 31536000 - d/365 = 734021 d - h/24 = (y - k) × 365 - h/24 = 190 h - m/60 = (d - k) × 24 - m/60 = 2 m - s/60 = (h - k) × 60 - s/60 = 47 s - cs/100 = (m - k) × 60 - cs/100 = 55 734021 years 190 days 02:47:55 734021 years 6 months 10 days 02:47:55
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  • QUESTION 23.

    TRIGONOMETRY WITHOUT EQUATIONS.

    θ = s/60 × 2π

    y = sin(θ) = -0.30901699437494

    x = cos(θ) = 0.95105651629515

    a. Solve for θ and s.

    sin^-1(y) = θ - 2π = -π/10

    θ = 2π + sin^-1(y) = 19π/10

    cos^-1(x) = 2π - θ = π/10

    θ = 2π - cos^-1(x) = 19π/10

    19π/10 = s/60 × 2π

    19/20 = s/60

    s = 19/20 × 60 = 57

    b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 57/60 = 19/20

    k = m - s/60 = 44

    m = k + s/60 = 899/20

    m/60 = 899/1200

    k = h - m/60 = 21

    h = k + m/60 = 26099/1200

    h/24 = 26099/28800

    k = d - h/24 = 15

    d = k + h/24 = 458099/28800

    d/30 = 458099/864000

    k = M - d/30 = 5

    M = k + d/30 = 4778099/864000

    M/(73/6) = 4778099/10512000

    k = y - M/(73/6) = 13423

    y = k + M/(73/6) = 141107354099/10512000

    c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B.

    T^1 = √(A/B) = 10^12 ps

    T^-1 = √(B/A) = 10^-12 Ts

    T^2 = A/B = 10^24 ps²

    T^-2 = B/A = 10^-24 Ts²

    t = A/T^1 = y × 31536000 = 4.23322062297 × 10^11 s

    A = tT^1 = X × 10^11 × 10^12 = X × 10^23 ps

    B = A/T^2 = X × 10^23 / 10^24 = X × 10^-1 Ts

    d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending.

    M/(73/6) = y - k = 4778099/10512000

    M = (y - k) × (73/6) = 4778099/864000

    k = M - d/30 = 5

    d/30 = M - k = 458099/864000

    d = (M - k) × 30 = 458099/28800

    k = d - h/24 = 15

    h/24 = d - k = 26099/28800

    h = (d - k) × 24 = 26099/1200

    k = h - m/60 = 21

    m/60 = h - k = 899/1200

    m = (h - k) × 60 = 899/20

    k = m - s/60 = 44

    s/60 = m - k = 19/20

    s = (m - k) × 60 = 57

    e. Check months with days.

    y - d/365 = t / 31536000 - d/365 = 13423

    d - h/24 = (y - k) × 365 - h/24 = 165

    h - m/60 = (d - k) × 24 - m/60 = 21

    m - s/60 = (h - k) × 60 - s/60 = 44

    s - cs/100 = (m - k) × 60 - cs/100 = 57

    13423 years 165 days 21:44:57

    13423 years 5 months 15 days 21:44:57
    QUESTION 23. TRIGONOMETRY WITHOUT EQUATIONS. θ = s/60 × 2π y = sin(θ) = -0.30901699437494 x = cos(θ) = 0.95105651629515 a. Solve for θ and s. sin^-1(y) = θ - 2π = -π/10 θ = 2π + sin^-1(y) = 19π/10 cos^-1(x) = 2π - θ = π/10 θ = 2π - cos^-1(x) = 19π/10 19π/10 = s/60 × 2π 19/20 = s/60 s = 19/20 × 60 = 57 b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 57/60 = 19/20 k = m - s/60 = 44 m = k + s/60 = 899/20 m/60 = 899/1200 k = h - m/60 = 21 h = k + m/60 = 26099/1200 h/24 = 26099/28800 k = d - h/24 = 15 d = k + h/24 = 458099/28800 d/30 = 458099/864000 k = M - d/30 = 5 M = k + d/30 = 4778099/864000 M/(73/6) = 4778099/10512000 k = y - M/(73/6) = 13423 y = k + M/(73/6) = 141107354099/10512000 c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B. T^1 = √(A/B) = 10^12 ps T^-1 = √(B/A) = 10^-12 Ts T^2 = A/B = 10^24 ps² T^-2 = B/A = 10^-24 Ts² t = A/T^1 = y × 31536000 = 4.23322062297 × 10^11 s A = tT^1 = X × 10^11 × 10^12 = X × 10^23 ps B = A/T^2 = X × 10^23 / 10^24 = X × 10^-1 Ts d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending. M/(73/6) = y - k = 4778099/10512000 M = (y - k) × (73/6) = 4778099/864000 k = M - d/30 = 5 d/30 = M - k = 458099/864000 d = (M - k) × 30 = 458099/28800 k = d - h/24 = 15 h/24 = d - k = 26099/28800 h = (d - k) × 24 = 26099/1200 k = h - m/60 = 21 m/60 = h - k = 899/1200 m = (h - k) × 60 = 899/20 k = m - s/60 = 44 s/60 = m - k = 19/20 s = (m - k) × 60 = 57 e. Check months with days. y - d/365 = t / 31536000 - d/365 = 13423 d - h/24 = (y - k) × 365 - h/24 = 165 h - m/60 = (d - k) × 24 - m/60 = 21 m - s/60 = (h - k) × 60 - s/60 = 44 s - cs/100 = (m - k) × 60 - cs/100 = 57 13423 years 165 days 21:44:57 13423 years 5 months 15 days 21:44:57
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  • QUESTION 22.

    TRIGONOMETRIC EQUATIONS.

    θ = s/60 × 2π

    3sin^3(θ)/4 = -3/32

    6cos^2(θ)/12 = 3/8

    a. Solve for θ and s.

    3sin^3(θ) = -3/8

    sin^3(θ) = -1/8

    sin(θ) = ³√(-1/8) = -1/2

    sin^-1(-1/2) = θ - 2π = -π/6

    θ = 2π + sin^-1(-1/2) = 11π/6

    6cos^2(θ) = 9/2

    cos^2(θ) = 3/4

    cos(θ) = √(3/4) = √3/2

    cos^-1(√3/2) = 2π - θ = π/6

    θ = 2π - cos^-1(√3/2) = 11π/6

    11π/6 = s/60 × 2π

    11/12 = s/60

    s = 11/12 × 60 = 55

    b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 55/60 = 11/12

    k = m - s/60 = 17

    m = k + s/60 = 215/12

    m/60 = 43/144

    k = h - m/60 = 9

    h = k + m/60 = 1339/144

    h/24 = 1339/3456

    k = d - h/24 = 18

    d = k + h/24 = 63547/3456

    d/30 = 63547/103680

    k = M - d/30 = 3

    M = k + d/30 = 374587/103680

    M/(73/6) = 374587/1261440

    k = y - M/(73/6) = 43179

    y = k + M/(73/6) = 54468092347/1261440

    c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B.

    T^1 = √(A/B) = 10^6 μs

    T^-1 = √(B/A) = 10^-6 Ms

    T^2 = A/B = 10^12 μs²

    T^-2 = B/A = 10^-12 Ms²

    t = A/T^1 = y × 31536000 = 1.361702308675 × 10^12 s

    A = tT^1 = X × 10^12 × 10^6 = X × 10^18 μs

    B = A/T^2 = X × 10^18 / 10^12 = X × 10^6 Ms

    d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending.

    M/(73/6) = y - k = 374587/1261440

    M = (y - k) × (73/6) = 374587/103680

    k = M - d/30 = 3

    d/30 = M - k = 63547/103680

    d = (M - k) × 30 = 63547/3456

    k = d - h/24 = 18

    h/24 = d - k = 1339/3456

    h = (d - k) × 24 = 1339/144

    k = h - m/60 = 9

    m/60 = h - k = 43/144

    m = (h - k) × 60 = 215/12

    k = m - s/60 = 17

    s/60 = m - k = 11/12

    s = (m - k) × 60 = 55

    e. Check months with days.

    y - d/365 = t / 31536000 - d/365 = 43179

    d - h/24 = (y - k) × 365 - h/24 = 108

    h - m/60 = (d - k) × 24 - m/60 = 9

    m - s/60 = (h - k) × 60 - s/60 = 17

    s - cs/100 = (m - k) × 60 - cs/100 = 55

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 1361702308675 / 31536000 - d/365 = 43179

    d - h/24 = (43,179.2969518962 - 43179) × 365 - h/24 = 108

    h - m/60 = (108.387442129629 - 108) × 24 - m/60 = 9

    m - s/60 = (9.29861111111111 - 9) × 60 - s/60 = 17

    s - cs/100 = (17.9166666666666 - 17) × 60 - cs/100 = 55

    43179 years 108 days 09:17:55

    43179 years 3 months 18 days 09:17:55
    QUESTION 22. TRIGONOMETRIC EQUATIONS. θ = s/60 × 2π 3sin^3(θ)/4 = -3/32 6cos^2(θ)/12 = 3/8 a. Solve for θ and s. 3sin^3(θ) = -3/8 sin^3(θ) = -1/8 sin(θ) = ³√(-1/8) = -1/2 sin^-1(-1/2) = θ - 2π = -π/6 θ = 2π + sin^-1(-1/2) = 11π/6 6cos^2(θ) = 9/2 cos^2(θ) = 3/4 cos(θ) = √(3/4) = √3/2 cos^-1(√3/2) = 2π - θ = π/6 θ = 2π - cos^-1(√3/2) = 11π/6 11π/6 = s/60 × 2π 11/12 = s/60 s = 11/12 × 60 = 55 b. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 55/60 = 11/12 k = m - s/60 = 17 m = k + s/60 = 215/12 m/60 = 43/144 k = h - m/60 = 9 h = k + m/60 = 1339/144 h/24 = 1339/3456 k = d - h/24 = 18 d = k + h/24 = 63547/3456 d/30 = 63547/103680 k = M - d/30 = 3 M = k + d/30 = 374587/103680 M/(73/6) = 374587/1261440 k = y - M/(73/6) = 43179 y = k + M/(73/6) = 54468092347/1261440 c. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2, also workout t, A and B. T^1 = √(A/B) = 10^6 μs T^-1 = √(B/A) = 10^-6 Ms T^2 = A/B = 10^12 μs² T^-2 = B/A = 10^-12 Ms² t = A/T^1 = y × 31536000 = 1.361702308675 × 10^12 s A = tT^1 = X × 10^12 × 10^6 = X × 10^18 μs B = A/T^2 = X × 10^18 / 10^12 = X × 10^6 Ms d. Although you have done it forward or ascending, undo or reverse-workout the integers and decimals of the different time units backward or descending. M/(73/6) = y - k = 374587/1261440 M = (y - k) × (73/6) = 374587/103680 k = M - d/30 = 3 d/30 = M - k = 63547/103680 d = (M - k) × 30 = 63547/3456 k = d - h/24 = 18 h/24 = d - k = 1339/3456 h = (d - k) × 24 = 1339/144 k = h - m/60 = 9 m/60 = h - k = 43/144 m = (h - k) × 60 = 215/12 k = m - s/60 = 17 s/60 = m - k = 11/12 s = (m - k) × 60 = 55 e. Check months with days. y - d/365 = t / 31536000 - d/365 = 43179 d - h/24 = (y - k) × 365 - h/24 = 108 h - m/60 = (d - k) × 24 - m/60 = 9 m - s/60 = (h - k) × 60 - s/60 = 17 s - cs/100 = (m - k) × 60 - cs/100 = 55 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 1361702308675 / 31536000 - d/365 = 43179 d - h/24 = (43,179.2969518962 - 43179) × 365 - h/24 = 108 h - m/60 = (108.387442129629 - 108) × 24 - m/60 = 9 m - s/60 = (9.29861111111111 - 9) × 60 - s/60 = 17 s - cs/100 = (17.9166666666666 - 17) × 60 - cs/100 = 55 43179 years 108 days 09:17:55 43179 years 3 months 18 days 09:17:55
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  • TRIGONOMETRY AND RADIANS

    When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful.

    However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s).

    For example:

    0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

    0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

    0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

    0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

    0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

    0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

    0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

    0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

    0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

    0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

    0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s.

    However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5.

    For example:

    0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s.

    0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s.

    0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s.

    0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s.

    0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s.

    0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s.

    0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s.

    0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s.

    0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s.

    0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s.

    0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s.

    As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6).

    For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it.

    QUADRANTS

    The following is the formula for attaining θ in the 4 quadrants of the unit circle:

    Q1.

    sin^-1(y) = θ

    cos^-1(x) = θ

    Q2.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = θ

    Q3.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Q4.

    sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    QUESTION 21:

    (TRIGONOMETRIC EQUATIONS).

    θ = s/60 × 2π

    2sin^3(θ) = (-3√3)/4

    3cos^3(θ) + 6 = 45/8

    a. Solve for θ.

    sin^3(θ) = (-3√3)/8

    sin(θ) = ³√((-3√3)/8) = -√(3)/2

    sin^-1(-√(3)/2) = π - θ = -π/3

    θ = π - sin^-1(-√(3)/2) = 4π/3

    3cos^3(θ) = 45/8 - 6 = -3/8

    cos(θ)^3 = -1/8

    cos(θ) = ³√(-1/8) = -1/2

    cos^-1(-1/2) = 2π - θ = 2π/3

    θ = 2π - cos^-1(-1/2) = 4π/3

    b. Solve for s.

    4π/3 = s/60 × 2π

    2/3 = s/60

    s = 2/3 × 60 = 40

    c. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 2/3

    k = m - s/60 = 43

    m = k + s/60 = 131/3

    m/60 = 131/180

    k = h - m/60 = 15

    h = k + m/60 = 2831/180

    h/24 = 2831/4320

    k = d - h/24 = 17

    d = k + h/24 = 76271/4320

    d/30 = 76271/129600

    k = M - d/30 = 8

    M = k + d/30 = 1113071/129600

    M/(73/6) = 1113071/1576800

    k = y - M/(73/6) = 11380

    y = k + M/(73/6) = 17945097071/1576800

    d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ.

    y = sin(M(73/6) × 2π) = -0.96186457013315

    x = cos(M(73/6) × 2π) = -0.27352613901153

    Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here:

    Natural Scientific Calculator (NSC).

    sin^-1(y) = π - θ = -324671π/788400

    θ = π - sin^-1(y) = 1113071π/1576800

    cos^-1(x) = 2π - θ = 463729π/788400

    θ = 2π - cos^-1(x) = 1113071π/1576800

    e. Although you have already done it workout M/(73/6), M and k.

    M/(73/6) = θ/2π = 1113071/1576800

    M = M/(73/6) × (73/6) = 1113071/129600

    k = M - d/30 = 8

    f. Although you have already done it from (s) upwards, reverse-workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.

    d/30 = M - k = 76271/129600

    d = (M - k) × 30 = 76271/4320

    k = d - h/24 = 17

    h/24 = d - k = 2831/4320

    h = (d - k) × 24 = 2831/180

    k = h - m/60 = 15

    m/60 = h - k = 131/180

    m = (h - k) × 60 = 131/3

    k = m - s/60 = 43

    s/60 = m - k = 2/3

    s = (m - k) × 60 = 40

    t = 11380 years 8 months 17 days 15:43:40

    g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?

    T^1 = √(A/B) = 10^3 ms

    T^-1 = √(B/A) = 10^-3 ks

    T^2 = A/B = 10^6 ms²

    T^-2 = B/A = 10^-6 ks²

    h. Workout the values of t, A and B?

    t = A/T^1 = y × 31536000 = 3.58901941420 × 10^11 s

    A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms

    B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks

    i. Check months and days.

    y - d/365 = t / 31536000 - d/365 = 11380

    d - h/24 = d/365 × 365 - h/24 = 257

    h - m/60 = h/24 × 24 - m/60 = 15

    m - s/60 = m/60 × 60 - s/60 = 43

    s - cs/100 = s/60 × 60 - cs/100 = 40

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 358901941420 / 31536000 - d/365 = 11380

    d - h/24 = (11,380.7059049974 - 11380) × 365 - h/24 = 257

    h - m/60 = (257.655324074074 - 257) × 24 - m/60 = 15

    m - s/60 = (15.7277777777777 - 15) × 60 - s/60 = 43

    s - cs/100 = (43.6666666666666 - 43) × 60 - cs/100 = 40

    t = 11380 years 257 days 15:43:40
    TRIGONOMETRY AND RADIANS When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful. However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s). For example: 0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s. 0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s. 0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s. 0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s. 0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s. 0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s. 0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s. 0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s. 0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s. 0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s. 0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s. However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5. For example: 0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s. 0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s. 0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s. 0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s. 0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s. 0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s. 0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s. 0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s. 0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s. 0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s. 0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s. As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6). For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it. QUADRANTS The following is the formula for attaining θ in the 4 quadrants of the unit circle: Q1. sin^-1(y) = θ cos^-1(x) = θ Q2. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = θ Q3. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Q4. sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) QUESTION 21: (TRIGONOMETRIC EQUATIONS). θ = s/60 × 2π 2sin^3(θ) = (-3√3)/4 3cos^3(θ) + 6 = 45/8 a. Solve for θ. sin^3(θ) = (-3√3)/8 sin(θ) = ³√((-3√3)/8) = -√(3)/2 sin^-1(-√(3)/2) = π - θ = -π/3 θ = π - sin^-1(-√(3)/2) = 4π/3 3cos^3(θ) = 45/8 - 6 = -3/8 cos(θ)^3 = -1/8 cos(θ) = ³√(-1/8) = -1/2 cos^-1(-1/2) = 2π - θ = 2π/3 θ = 2π - cos^-1(-1/2) = 4π/3 b. Solve for s. 4π/3 = s/60 × 2π 2/3 = s/60 s = 2/3 × 60 = 40 c. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 2/3 k = m - s/60 = 43 m = k + s/60 = 131/3 m/60 = 131/180 k = h - m/60 = 15 h = k + m/60 = 2831/180 h/24 = 2831/4320 k = d - h/24 = 17 d = k + h/24 = 76271/4320 d/30 = 76271/129600 k = M - d/30 = 8 M = k + d/30 = 1113071/129600 M/(73/6) = 1113071/1576800 k = y - M/(73/6) = 11380 y = k + M/(73/6) = 17945097071/1576800 d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ. y = sin(M(73/6) × 2π) = -0.96186457013315 x = cos(M(73/6) × 2π) = -0.27352613901153 Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here: Natural Scientific Calculator (NSC). sin^-1(y) = π - θ = -324671π/788400 θ = π - sin^-1(y) = 1113071π/1576800 cos^-1(x) = 2π - θ = 463729π/788400 θ = 2π - cos^-1(x) = 1113071π/1576800 e. Although you have already done it workout M/(73/6), M and k. M/(73/6) = θ/2π = 1113071/1576800 M = M/(73/6) × (73/6) = 1113071/129600 k = M - d/30 = 8 f. Although you have already done it from (s) upwards, reverse-workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order. d/30 = M - k = 76271/129600 d = (M - k) × 30 = 76271/4320 k = d - h/24 = 17 h/24 = d - k = 2831/4320 h = (d - k) × 24 = 2831/180 k = h - m/60 = 15 m/60 = h - k = 131/180 m = (h - k) × 60 = 131/3 k = m - s/60 = 43 s/60 = m - k = 2/3 s = (m - k) × 60 = 40 t = 11380 years 8 months 17 days 15:43:40 g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2? T^1 = √(A/B) = 10^3 ms T^-1 = √(B/A) = 10^-3 ks T^2 = A/B = 10^6 ms² T^-2 = B/A = 10^-6 ks² h. Workout the values of t, A and B? t = A/T^1 = y × 31536000 = 3.58901941420 × 10^11 s A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks i. Check months and days. y - d/365 = t / 31536000 - d/365 = 11380 d - h/24 = d/365 × 365 - h/24 = 257 h - m/60 = h/24 × 24 - m/60 = 15 m - s/60 = m/60 × 60 - s/60 = 43 s - cs/100 = s/60 × 60 - cs/100 = 40 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 358901941420 / 31536000 - d/365 = 11380 d - h/24 = (11,380.7059049974 - 11380) × 365 - h/24 = 257 h - m/60 = (257.655324074074 - 257) × 24 - m/60 = 15 m - s/60 = (15.7277777777777 - 15) × 60 - s/60 = 43 s - cs/100 = (43.6666666666666 - 43) × 60 - cs/100 = 40 t = 11380 years 257 days 15:43:40
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  • TRIGONOMETRY AND RADIANS

    When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful.

    However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s).

    For example:

    0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

    0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

    0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

    0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

    0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

    0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

    0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

    0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

    0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

    0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

    0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s.

    However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5.

    For example:

    0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s.

    0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s.

    0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s.

    0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s.

    0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s.

    0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s.

    0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s.

    0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s.

    0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s.

    0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s.

    0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s.

    As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6).

    For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it.

    QUADRANTS

    The following is the formula for attaining θ in the 4 quadrants of the unit circle:

    Q1.

    sin^-1(y) = θ

    cos^-1(x) = θ

    Q2.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = θ

    Q3.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Q4.

    sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    QUESTION 20:

    (TRIGONOMETRIC EQUATIONS).

    θ = s/60 × 2π

    sin^2(θ) + 2 = 11/4

    2cos(θ) + 1 = 0

    a. Solve for θ.

    sin^2(θ) =11/4 - 2 = 3/4

    sin(θ) = √(3/4) = √(3)/2

    sin^-1(√(3)/2) = π/3

    2cos(θ) = - 1

    cos(θ) = -1/2

    cos^-1(-1/2) = 2π/3

    θ = 2π/3

    b. Solve for s.

    2π/3 = s/60 × 2π

    1/3 = s/60

    s = 1/3 × 60 = 20

    c. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years then add k + M/(73/6) to get y.

    s/60 = 1/3

    m = k + s/60 = 49/3

    k = m - s/60 = 16

    m/60 = 49/180

    h = k + m/60 = 3829/180

    k = h - m/60 = 21

    h/24 = 3829/4320

    d = k + h/24 = 68629/4320

    k = d - h/24 = 15

    d/30 = 68629/129600

    M = k + d/30 = 846229/129600

    k = M - d/30 = 6

    M/(73/6) = 846229/1576800

    y = k + M/(73/6) = 103629719029/1576800

    k = y - M/(73/6) = 65721

    d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ.

    y = sin(M(73/6) × 2π) = 0.05881902090299

    x = cos(M(73/6) × 2π) = 0.99826866262545

    Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here:

    https://8uvp.app.link/FeZPwAXF1V

    sin^-1(y) = θ = 846229π/788400

    cos^-1(x) = θ = 846229π/788400

    e. Although you have already done it workout M/(73/6), M and k.

    M/(73/6) = θ/2π = 846229/1576800

    M = M/(73/6) × (73/6) = 846229/129600

    k = M - d/30 = 6

    f. Although you have already done it from (s), workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.

    d/30 = M - k = 68629/129600

    d = (M - k) × 30 = 68629/4320

    k = d - h/24 = 15

    h/24 = d - k = 3829/4320

    h = (d - k) × 24 = 3829/180

    k = h - m/60 = 21

    m/60 = h - k = 49/180

    m = (h - k) × 60 = 49/3

    k = m - s/60 = 16

    s/60 = m - k = 1/3

    s = (m - k) × 60 = 20

    t = 65721 years 6 months 15 days 21:16:20

    g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?

    T^1 = √(A/B) = 10^15 fs

    T^-1 = √(B/A) = 10^-15 Ps

    T^2 = A/B = 10^30 fs²

    T^-2 = B/A = 10^-30 Ps²

    h. Workout the values of t, A and B?

    t = A/T^1 = y × 31536000 = 2.072594380580 × 10^12 s

    A = tT^1 = X × 10^12 × 10^15 = X × 10^27 fs

    B = A/T^2 = X × 10^27 / 10^30 = X × 10^-3 Ps

    i. Check months and days.

    y - d/365 = t / 31536000 - d/365 = 65721

    d - h/24 = d/365 × 365 - h/24 = 195

    h - m/60 = h/24 × 24 - m/60 = 21

    m - s/60 = m/60 × 60 - s/60 = 16

    s - cs/100 = s/60 × 60 - cs/100 = 20

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 2072594380580 / 31536000 - d/365 = 65721

    d - h/24 = (65721.5366749112 - 65721) × 365 - h/24 = 195

    h - m/60 = (195.886342592592 - 195) × 24 - m/60 = 21

    m - s/60 = (21.2722222222222 - 21) × 60 - s/60 = 16

    s - cs/100 = (16.3333333333333 - 16) × 60 - cs/100 = 20

    t = 65721 years 195 days 21:16:20
    TRIGONOMETRY AND RADIANS When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful. However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s). For example: 0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s. 0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s. 0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s. 0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s. 0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s. 0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s. 0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s. 0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s. 0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s. 0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s. 0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s. However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5. For example: 0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s. 0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s. 0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s. 0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s. 0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s. 0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s. 0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s. 0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s. 0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s. 0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s. 0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s. As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6). For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it. QUADRANTS The following is the formula for attaining θ in the 4 quadrants of the unit circle: Q1. sin^-1(y) = θ cos^-1(x) = θ Q2. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = θ Q3. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Q4. sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) QUESTION 20: (TRIGONOMETRIC EQUATIONS). θ = s/60 × 2π sin^2(θ) + 2 = 11/4 2cos(θ) + 1 = 0 a. Solve for θ. sin^2(θ) =11/4 - 2 = 3/4 sin(θ) = √(3/4) = √(3)/2 sin^-1(√(3)/2) = π/3 2cos(θ) = - 1 cos(θ) = -1/2 cos^-1(-1/2) = 2π/3 θ = 2π/3 b. Solve for s. 2π/3 = s/60 × 2π 1/3 = s/60 s = 1/3 × 60 = 20 c. Re-divide s by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years then add k + M/(73/6) to get y. s/60 = 1/3 m = k + s/60 = 49/3 k = m - s/60 = 16 m/60 = 49/180 h = k + m/60 = 3829/180 k = h - m/60 = 21 h/24 = 3829/4320 d = k + h/24 = 68629/4320 k = d - h/24 = 15 d/30 = 68629/129600 M = k + d/30 = 846229/129600 k = M - d/30 = 6 M/(73/6) = 846229/1576800 y = k + M/(73/6) = 103629719029/1576800 k = y - M/(73/6) = 65721 d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ. y = sin(M(73/6) × 2π) = 0.05881902090299 x = cos(M(73/6) × 2π) = 0.99826866262545 Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here: https://8uvp.app.link/FeZPwAXF1V sin^-1(y) = θ = 846229π/788400 cos^-1(x) = θ = 846229π/788400 e. Although you have already done it workout M/(73/6), M and k. M/(73/6) = θ/2π = 846229/1576800 M = M/(73/6) × (73/6) = 846229/129600 k = M - d/30 = 6 f. Although you have already done it from (s), workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order. d/30 = M - k = 68629/129600 d = (M - k) × 30 = 68629/4320 k = d - h/24 = 15 h/24 = d - k = 3829/4320 h = (d - k) × 24 = 3829/180 k = h - m/60 = 21 m/60 = h - k = 49/180 m = (h - k) × 60 = 49/3 k = m - s/60 = 16 s/60 = m - k = 1/3 s = (m - k) × 60 = 20 t = 65721 years 6 months 15 days 21:16:20 g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2? T^1 = √(A/B) = 10^15 fs T^-1 = √(B/A) = 10^-15 Ps T^2 = A/B = 10^30 fs² T^-2 = B/A = 10^-30 Ps² h. Workout the values of t, A and B? t = A/T^1 = y × 31536000 = 2.072594380580 × 10^12 s A = tT^1 = X × 10^12 × 10^15 = X × 10^27 fs B = A/T^2 = X × 10^27 / 10^30 = X × 10^-3 Ps i. Check months and days. y - d/365 = t / 31536000 - d/365 = 65721 d - h/24 = d/365 × 365 - h/24 = 195 h - m/60 = h/24 × 24 - m/60 = 21 m - s/60 = m/60 × 60 - s/60 = 16 s - cs/100 = s/60 × 60 - cs/100 = 20 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 2072594380580 / 31536000 - d/365 = 65721 d - h/24 = (65721.5366749112 - 65721) × 365 - h/24 = 195 h - m/60 = (195.886342592592 - 195) × 24 - m/60 = 21 m - s/60 = (21.2722222222222 - 21) × 60 - s/60 = 16 s - cs/100 = (16.3333333333333 - 16) × 60 - cs/100 = 20 t = 65721 years 195 days 21:16:20
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  • TRIGONOMETRY AND RADIANS

    When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful.

    However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s).

    For example:

    0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

    0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

    0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

    0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

    0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

    0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

    0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

    0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

    0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

    0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

    0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s.

    However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5.

    For example:

    0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s.

    0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s.

    0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s.

    0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s.

    0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s.

    0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s.

    0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s.

    0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s.

    0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s.

    0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s.

    0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s.

    As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6).

    For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it.

    QUADRANTS

    The following is the formula for attaining θ in the 4 quadrants of the unit circle:

    Q1.

    sin^-1(y) = θ

    cos^-1(x) = θ

    Q2.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = θ

    Q3.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Q4.

    sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    QUESTION 19:

    (CREATE YOUR OWN QUESTIONS).

    a. Choose any whole number between 1 and 59 for seconds (s) and divide it by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y.

    s/60 = 11/20

    m = k + s/60 = 531/20

    k = m - s/60 = 26

    m/60 = 177/400

    h = k + m/60 = 7377/400

    k = h - m/60 = 18

    h/24 = 2459/3200

    d = k + h/24 = 69659/3200

    k = d - h/24 = 21

    d/30 = 69659/96000

    M = k + d/30 = 1029659/96000

    k = M - d/30 = 10

    M/(73/6) = 1029659/1168000

    y = k + M/(73/6) = 67305861659/1168000

    k = y - M/(73/6) = 57624

    b. Create angle (θ) from M/(73/6) × 2π and get the sine and cosine.

    y = sin(M(73/6) × 2π) = -0.67738134317054

    x = cos(M(73/6) × 2π) = -0.73563205199642

    c. Although you already know it and the object is defeated, workout θ.

    Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here: https://8uvp.app.link/FeZPwAXF1V

    sin^-1(y) = θ - 2π = -138341π/584000

    θ = 2π + sin^-1(y) = 1029659π/584000

    cos^-1(x) = 2π - θ = 138341π/584000

    θ = 2π - cos^-1(x) = 1029659π/584000

    d. Although you have already done it workout M/(73/6), M and k.

    M/(73/6) = θ/2π = 1029659/1168000

    M = M/(73/6) × (73/6) = 1029659/96000

    k = M - d/30 = 10

    e. Although you have already done it from (s), workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.

    d/30 = M - k = 69659/96000

    d = (M - k) × 30 = 69659/3200

    k = d - h/24 = 21

    h/24 = d - k = 2459/3200

    h = (d - k) × 24 = 7377/400

    k = h - m/60 = 18

    m/60 = h - k = 177/400

    m = (h - k) × 60 = 531/20

    k = m - s/60 = 26

    s/60 = m - k = 11/20

    s = (m - k) × 60 = 33

    t = 57624 years 10 months 21 days 18:26:33

    e. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?

    T^1 = √(A/B) = 10^12 ps

    T^-1 = √(B/A) = 10^-12 Ts

    T^2 = A/B = 10^24 ps²

    T^-2 = B/A = 10^-24 Ts²

    g. Workout the values of t, A and B?

    t = A/T^1 = y × 31536000 = 1.817258264793 × 10^12 s

    A = tT^1 = X × 10^12 × 10^12 = X × 10^24 ps

    B = A/T^2 = X × 10^24 / 10^24 = X × 10^0 Ts

    h. Check months and days.

    y - d/365 = t / 31536000 - d/365 = 57624

    d - h/24 = d/365 × 365 - h/24 = 321

    h - m/60 = h/24 × 24 - m/60 = 18

    m - s/60 = m/60 × 60 - s/60 = 26

    s - cs/100 = s/60 × 60 - cs/100 = 33

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below such as y - M/(365/31) and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 1817258264793 / 31536000 - d/365 = 57624

    d - h/24 = (57624.8815573630 - 57624) × 365 - h/24 = 321

    h - m/60 = (321.7684375 - 321) × 24 - m/60 = 18

    m - s/60 = (18.4425 - 18) × 60 - s/60 = 26

    s - cs/100 = (26.55 - 26) × 60 - cs/100 = 33

    t = 57624 years 321 days 18:26:33
    TRIGONOMETRY AND RADIANS When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful. However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s). For example: 0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s. 0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s. 0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s. 0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s. 0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s. 0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s. 0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s. 0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s. 0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s. 0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s. 0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s. However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5. For example: 0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s. 0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s. 0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s. 0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s. 0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s. 0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s. 0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s. 0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s. 0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s. 0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s. 0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s. As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6). For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it. QUADRANTS The following is the formula for attaining θ in the 4 quadrants of the unit circle: Q1. sin^-1(y) = θ cos^-1(x) = θ Q2. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = θ Q3. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Q4. sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) QUESTION 19: (CREATE YOUR OWN QUESTIONS). a. Choose any whole number between 1 and 59 for seconds (s) and divide it by 60 to get s/60. Then choose k for minutes (m) and add k + s/60 to get m. Then divide m by 60 and choose k for hours (h) then add k + m/60 to get h. Then divide h by 24 and choose k for days (d) then add k + h/24 to get d. Then divide d by 30 and choose k for months (M) then add k + d/30 to get M. Then divide M by (73/6) and choose k for years (y) then add k + M/(73/6) to get y. s/60 = 11/20 m = k + s/60 = 531/20 k = m - s/60 = 26 m/60 = 177/400 h = k + m/60 = 7377/400 k = h - m/60 = 18 h/24 = 2459/3200 d = k + h/24 = 69659/3200 k = d - h/24 = 21 d/30 = 69659/96000 M = k + d/30 = 1029659/96000 k = M - d/30 = 10 M/(73/6) = 1029659/1168000 y = k + M/(73/6) = 67305861659/1168000 k = y - M/(73/6) = 57624 b. Create angle (θ) from M/(73/6) × 2π and get the sine and cosine. y = sin(M(73/6) × 2π) = -0.67738134317054 x = cos(M(73/6) × 2π) = -0.73563205199642 c. Although you already know it and the object is defeated, workout θ. Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here: https://8uvp.app.link/FeZPwAXF1V sin^-1(y) = θ - 2π = -138341π/584000 θ = 2π + sin^-1(y) = 1029659π/584000 cos^-1(x) = 2π - θ = 138341π/584000 θ = 2π - cos^-1(x) = 1029659π/584000 d. Although you have already done it workout M/(73/6), M and k. M/(73/6) = θ/2π = 1029659/1168000 M = M/(73/6) × (73/6) = 1029659/96000 k = M - d/30 = 10 e. Although you have already done it from (s), workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order. d/30 = M - k = 69659/96000 d = (M - k) × 30 = 69659/3200 k = d - h/24 = 21 h/24 = d - k = 2459/3200 h = (d - k) × 24 = 7377/400 k = h - m/60 = 18 m/60 = h - k = 177/400 m = (h - k) × 60 = 531/20 k = m - s/60 = 26 s/60 = m - k = 11/20 s = (m - k) × 60 = 33 t = 57624 years 10 months 21 days 18:26:33 e. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2? T^1 = √(A/B) = 10^12 ps T^-1 = √(B/A) = 10^-12 Ts T^2 = A/B = 10^24 ps² T^-2 = B/A = 10^-24 Ts² g. Workout the values of t, A and B? t = A/T^1 = y × 31536000 = 1.817258264793 × 10^12 s A = tT^1 = X × 10^12 × 10^12 = X × 10^24 ps B = A/T^2 = X × 10^24 / 10^24 = X × 10^0 Ts h. Check months and days. y - d/365 = t / 31536000 - d/365 = 57624 d - h/24 = d/365 × 365 - h/24 = 321 h - m/60 = h/24 × 24 - m/60 = 18 m - s/60 = m/60 × 60 - s/60 = 26 s - cs/100 = s/60 × 60 - cs/100 = 33 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below such as y - M/(365/31) and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 1817258264793 / 31536000 - d/365 = 57624 d - h/24 = (57624.8815573630 - 57624) × 365 - h/24 = 321 h - m/60 = (321.7684375 - 321) × 24 - m/60 = 18 m - s/60 = (18.4425 - 18) × 60 - s/60 = 26 s - cs/100 = (26.55 - 26) × 60 - cs/100 = 33 t = 57624 years 321 days 18:26:33
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  • TRIGONOMETRY AND RADIANS

    When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful.

    However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s).

    For example:

    0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

    0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

    0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

    0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

    0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

    0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

    0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

    0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

    0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

    0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

    0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s.

    However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5.

    For example:

    0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s.

    0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s.

    0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s.

    0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s.

    0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s.

    0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s.

    0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s.

    0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s.

    0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s.

    0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s.

    0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s.

    As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6).

    For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it.

    QUADRANTS

    The following is the formula for attaining θ in the 4 quadrants of the unit circle:

    Q1.

    sin^-1(y) = θ

    cos^-1(x) = θ

    Q2.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = θ

    Q3.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Q4.

    sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Question 18:

    (TRIGONOMETRY AND RADIANS).

    T^1 = √(A/B) = 10^18 as

    k = y - M/(73/6) = 57892

    sin(M(73/6) × 2π) = -0.5278455119451

    cos(M(73/6) × 2π) = -0.8493404002633

    a. Workout θ.

    b. Workout M/(73/6), M and k.

    c. Workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds.

    d. What is the total value of y?

    e. What are the magnitudes of T^-1, T^2 and T^-2?

    f. What are the values of t, A and B?

    g. Check the days and months.

    Note: to figure out a fraction radian from a decimal radian, divide the decimal radian by 2π then convert the resulting decimal into a fraction.

    For example:

    0.6471680866395 / 2π = 0.103 = 103/1000

    Then multiply the fraction by 2π.

    radian = 103π/500

    a.

    sin^-1(-0.5278455119451) = π - θ = -177π/1000

    θ = π - (-177π/1000) = 1177π/1000

    cos^-1(-0.8493404002633) = 2π - θ = 823π/1000

    θ = 2π - 823π/1000 = 1177π/1000

    b.

    M/(73/6) = θ/2π = (1177π/1000)/2π = 1177/2000

    M = M/(73/6) × (73/6) = 85921/12000

    k = M - d/30 = 7

    c.

    d/30 = M - k = 1921/12000

    d = (M - k) × 30 = 1921/400

    k = d - h/24 = 4

    h/24 = d - k = 321/400

    h = (d - k) × 24 = 963/50

    k = h - m/60 = 19

    m/60 = h - k = 13/50

    m = (h - k) × 60 = 78/5

    k = m - s/60 = 15

    s/60 = m - k = 3/5

    s = (m - k) × 60 = 36

    t = 57892 years 7 months 4 days 19:15:36

    d.

    y = k + M/(73/6) = 115785177/2000

    e.

    T^-1 = √(B/A) = 10^-18 Es

    T^2 = A/B = 10^36 as²

    T^-2 = B/A = 10^-36 Es²

    f.

    t = A/T^1 = y × 31536000 = 1.825700670936 × 10^12 s

    A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as

    B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es

    g.

    y - d/365 = t / 31536000 - d/365 = 57892

    d - h/24 = d/365 × 365 - h/24 = 214

    h - m/60 = h/24 × 24 - m/60 = 19

    m - s/60 = m/60 × 60 - s/60 = 15

    s - cs/100 = s/60 × 60 - cs/100 = 36

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 1825700670936 / 31536000 - d/365 = 57892

    d - h/24 = (57892.5885 - 57892) × 365 - h/24 = 214

    h - m/60 = (214.802499999278 - 214) × 24 - m/60 = 19

    m - s/60 = (19.2599999826634 - 19) × 60 - s/60 = 15

    s - cs/100 = (15.5999989598058 - 15) × 60 - cs/100 = 36

    t = 57892 years 214 days 19:15:36
    TRIGONOMETRY AND RADIANS When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful. However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s). For example: 0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s. 0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s. 0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s. 0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s. 0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s. 0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s. 0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s. 0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s. 0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s. 0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s. 0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s. However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5. For example: 0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s. 0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s. 0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s. 0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s. 0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s. 0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s. 0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s. 0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s. 0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s. 0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s. 0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s. As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6). For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it. QUADRANTS The following is the formula for attaining θ in the 4 quadrants of the unit circle: Q1. sin^-1(y) = θ cos^-1(x) = θ Q2. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = θ Q3. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Q4. sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Question 18: (TRIGONOMETRY AND RADIANS). T^1 = √(A/B) = 10^18 as k = y - M/(73/6) = 57892 sin(M(73/6) × 2π) = -0.5278455119451 cos(M(73/6) × 2π) = -0.8493404002633 a. Workout θ. b. Workout M/(73/6), M and k. c. Workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds. d. What is the total value of y? e. What are the magnitudes of T^-1, T^2 and T^-2? f. What are the values of t, A and B? g. Check the days and months. Note: to figure out a fraction radian from a decimal radian, divide the decimal radian by 2π then convert the resulting decimal into a fraction. For example: 0.6471680866395 / 2π = 0.103 = 103/1000 Then multiply the fraction by 2π. radian = 103π/500 a. sin^-1(-0.5278455119451) = π - θ = -177π/1000 θ = π - (-177π/1000) = 1177π/1000 cos^-1(-0.8493404002633) = 2π - θ = 823π/1000 θ = 2π - 823π/1000 = 1177π/1000 b. M/(73/6) = θ/2π = (1177π/1000)/2π = 1177/2000 M = M/(73/6) × (73/6) = 85921/12000 k = M - d/30 = 7 c. d/30 = M - k = 1921/12000 d = (M - k) × 30 = 1921/400 k = d - h/24 = 4 h/24 = d - k = 321/400 h = (d - k) × 24 = 963/50 k = h - m/60 = 19 m/60 = h - k = 13/50 m = (h - k) × 60 = 78/5 k = m - s/60 = 15 s/60 = m - k = 3/5 s = (m - k) × 60 = 36 t = 57892 years 7 months 4 days 19:15:36 d. y = k + M/(73/6) = 115785177/2000 e. T^-1 = √(B/A) = 10^-18 Es T^2 = A/B = 10^36 as² T^-2 = B/A = 10^-36 Es² f. t = A/T^1 = y × 31536000 = 1.825700670936 × 10^12 s A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es g. y - d/365 = t / 31536000 - d/365 = 57892 d - h/24 = d/365 × 365 - h/24 = 214 h - m/60 = h/24 × 24 - m/60 = 19 m - s/60 = m/60 × 60 - s/60 = 15 s - cs/100 = s/60 × 60 - cs/100 = 36 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 1825700670936 / 31536000 - d/365 = 57892 d - h/24 = (57892.5885 - 57892) × 365 - h/24 = 214 h - m/60 = (214.802499999278 - 214) × 24 - m/60 = 19 m - s/60 = (19.2599999826634 - 19) × 60 - s/60 = 15 s - cs/100 = (15.5999989598058 - 15) × 60 - cs/100 = 36 t = 57892 years 214 days 19:15:36
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  • TRIGONOMETRY AND RADIANS

    To create your own trigonometry questions and to save you time, know that θ or (M/(73/6) × 360) should have 2 decimal places. Any less and there are 0 seconds, anymore and the maths is not beautiful.

    Note: 2 decimal places this means there are only 5 possible values for seconds (s).

    For example:

    0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

    0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

    0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

    0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

    0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

    0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

    0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

    0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

    0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

    0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

    0.10 or X.Y0= 60 s, E.G.: 156.40° will result in 60 s.

    Note: the Mathway app makes light work of fractions and radians, also a second scrap piece of paper is recommended.

    QUADRANTS

    The following is the formula for attaining θ in the 4 quadrants of the unit circle:

    Q1.

    sin^-1(y) = θ

    cos^-1(x) = θ

    Q2.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = θ

    Q3.

    sin^-1(y) = π - θ therefore θ = π - sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Q4.

    sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y)

    cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x)

    Question 17:

    (TRIGONOMETRY AND RADIANS).

    T^1 = √(A/B) = 10^21 zs

    k = y - M/(73/6) = 80723

    sin(M(73/6) × 2π) = -0.602929541689

    cos(M(73/6) × 2π) = -0.79779443953857

    a. Workout θ.

    b. Workout M/(73/6), M and k.

    c. Workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds.

    d. What is the total value of y?

    e. What are the magnitudes of T^-1, T^2 and T^-2?

    f. What are the values of t, A and B?

    g. Check the days and months.

    Note: to figure out a fraction radian from a decimal radian, divide the decimal radian by 2π then convert the resulting decimal into a fraction.

    For example:

    0.6471680866395 / 2π = 0.103 = 103/1000

    Then multiply the fraction by 2π.

    radian = 103π/500

    a.

    sin^-1(-0.602929541689) = θ - 2π = -103π/500

    θ = 2π - 103π/500 = 897π/500

    cos^-1(-0.79779443953857) = 2π - θ = 103π/500

    θ = 2π - 103π/500 = 897π/500

    b.

    M/(73/6) = θ/2π = (897π/500)/2π = 897/1000

    M = M/(73/6) × (73/6) = 21827/2000

    k = M - d/30 = 10

    c.

    d/30 = M - k = 1827/2000

    d = (M - k) × 30 = 5481/200

    k = d - h/24 = 27

    h/24 = d - k = 81/200

    h = (d - k) × 24 = 243/25

    k = h - m/60 = 9

    m/60 = h - k = 18/25

    m = (h - k) × 60 = 216/5

    k = m - s/60 = 43

    s/60 = m - k = 1/5

    s = (m - k) × 60 = 12

    t = 80723 years 10 months 27 days 09:43:12

    d.

    y = k + M/(73/6) = 80723897/1000

    e.

    T^-1 = √(B/A) = 10^-21 Zs

    T^2 = A/B = 10^42 zs²

    T^-2 = B/A = 10^-42 Zs²

    f.

    t = A/T^1 = y × 31536000 = 2.545708815792 × 10^12 s

    A = tT^1 = X × 10^12 × 10^21 = X × 10^33 zs

    B = A/T^2 = X × 10^33 / 10^42 = X × 10^-9 Zs

    g.

    y - d/365 = t / 31536000 - d/365 = 80723

    d - h/24 = d/365 × 365 - h/24 = 327

    h - m/60 = h/24 × 24 - m/60 = 9

    m - s/60 = m/60 × 60 - s/60 = 43

    s - cs/100 = s/60 × 60 - cs/100 = 12

    NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

    For example:

    Note: you do not need to write the below, it is all done on the calculator.

    y - d/365 = 2545708815792 / 31536000 - d/365 = 80723

    d - h/24 = (80723.897- 80723) × 365 - h/24 = 327

    h - m/60 = (327.40499999898 - 327) × 24 - m/60 = 9

    m - s/60 = (9.71999997552484 - 9) × 60 - s/60 = 43

    s - cs/100 = (43.1999985314906 - 43) × 60 - cs/100 = 12

    t = 80723 years 327 days 09:43:12
    TRIGONOMETRY AND RADIANS To create your own trigonometry questions and to save you time, know that θ or (M/(73/6) × 360) should have 2 decimal places. Any less and there are 0 seconds, anymore and the maths is not beautiful. Note: 2 decimal places this means there are only 5 possible values for seconds (s). For example: 0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s. 0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s. 0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s. 0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s. 0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s. 0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s. 0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s. 0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s. 0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s. 0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s. 0.10 or X.Y0= 60 s, E.G.: 156.40° will result in 60 s. Note: the Mathway app makes light work of fractions and radians, also a second scrap piece of paper is recommended. QUADRANTS The following is the formula for attaining θ in the 4 quadrants of the unit circle: Q1. sin^-1(y) = θ cos^-1(x) = θ Q2. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = θ Q3. sin^-1(y) = π - θ therefore θ = π - sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Q4. sin^-1(y) = θ - 2π therefore θ = 2π + sin^-1(y) cos^-1(x) = 2π - θ therefore θ = 2π - cos^-1(x) Question 17: (TRIGONOMETRY AND RADIANS). T^1 = √(A/B) = 10^21 zs k = y - M/(73/6) = 80723 sin(M(73/6) × 2π) = -0.602929541689 cos(M(73/6) × 2π) = -0.79779443953857 a. Workout θ. b. Workout M/(73/6), M and k. c. Workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds. d. What is the total value of y? e. What are the magnitudes of T^-1, T^2 and T^-2? f. What are the values of t, A and B? g. Check the days and months. Note: to figure out a fraction radian from a decimal radian, divide the decimal radian by 2π then convert the resulting decimal into a fraction. For example: 0.6471680866395 / 2π = 0.103 = 103/1000 Then multiply the fraction by 2π. radian = 103π/500 a. sin^-1(-0.602929541689) = θ - 2π = -103π/500 θ = 2π - 103π/500 = 897π/500 cos^-1(-0.79779443953857) = 2π - θ = 103π/500 θ = 2π - 103π/500 = 897π/500 b. M/(73/6) = θ/2π = (897π/500)/2π = 897/1000 M = M/(73/6) × (73/6) = 21827/2000 k = M - d/30 = 10 c. d/30 = M - k = 1827/2000 d = (M - k) × 30 = 5481/200 k = d - h/24 = 27 h/24 = d - k = 81/200 h = (d - k) × 24 = 243/25 k = h - m/60 = 9 m/60 = h - k = 18/25 m = (h - k) × 60 = 216/5 k = m - s/60 = 43 s/60 = m - k = 1/5 s = (m - k) × 60 = 12 t = 80723 years 10 months 27 days 09:43:12 d. y = k + M/(73/6) = 80723897/1000 e. T^-1 = √(B/A) = 10^-21 Zs T^2 = A/B = 10^42 zs² T^-2 = B/A = 10^-42 Zs² f. t = A/T^1 = y × 31536000 = 2.545708815792 × 10^12 s A = tT^1 = X × 10^12 × 10^21 = X × 10^33 zs B = A/T^2 = X × 10^33 / 10^42 = X × 10^-9 Zs g. y - d/365 = t / 31536000 - d/365 = 80723 d - h/24 = d/365 × 365 - h/24 = 327 h - m/60 = h/24 × 24 - m/60 = 9 m - s/60 = m/60 × 60 - s/60 = 43 s - cs/100 = s/60 × 60 - cs/100 = 12 NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator. For example: Note: you do not need to write the below, it is all done on the calculator. y - d/365 = 2545708815792 / 31536000 - d/365 = 80723 d - h/24 = (80723.897- 80723) × 365 - h/24 = 327 h - m/60 = (327.40499999898 - 327) × 24 - m/60 = 9 m - s/60 = (9.71999997552484 - 9) × 60 - s/60 = 43 s - cs/100 = (43.1999985314906 - 43) × 60 - cs/100 = 12 t = 80723 years 327 days 09:43:12
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