# Question 22: Trigonometric equations.

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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