# Question 24: Trigonometric equations. (Lower Palaeolithic).

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 acceptance and forgiveness. For a full understanding of this read https://primitiverelativity.com.

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 Paleolithic (or Lower Palaeolithic) is the earliest subdivision of the Paleolithic 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, spanning the Oldowan (“mode 1”) and Acheulean (“mode 2”) lithics industries.

https://en.wikipedia.org/wiki/Lower_Paleolithic Example of Neanderthal humans in Old stone age. Image by artist Zdeněk Burian (1905-1981). https://www.shorthistory.org/prehistory/palaeolithic-old-stone-age/