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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