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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