QUESTION 24.

TRIGONOMETRIC EQUATIONS (Lower Palaeolithic).

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.

Tip: choose a meaningful k for years (y), especially if you have sinned, such as the Lower Palaeolithic, 3 million to 300,000 years ago. For example, if you are a genocidal megalomaniac, this is a place and time where you can find relative innocence. For a full understanding of this read https://crimetravel.co.uk.

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

Hand written example:

The Lower Palaeolithic is the earliest subdivision of the Palaeolithic or Old Stone Age. It spans the time from around 3 million years ago when the first evidence for stone tool production and use by hominins appears in the current archaeological record, ^{}until around 300,000 years ago. Again, for example, if you are a genocidal megalomaniac, this is a place and time where you can find relative innocence. For a full understanding of this read https://crimetravel.co.uk.