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

Hand written example:

 

Author:

The law in one frame of reference or time period is not the law in another frame of reference or time period. The law changes over space and time. The law is not absolute. The law is relative. The law is flexible.

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