# Question 21: Trigonometric equations.

QUESTION 21.

When creating your own trigonometry and radians questions know that if we are being lazy and starting from an angle of M/(73/6) and we want particularly pretty maths or simple fractions then θ needs 2 decimal places. Any less and there are 0 seconds, anymore and the seconds are not a nice whole number. However, if you create the question from the ground up, that is starting with a nice whole number for the seconds and building up, then the maths is very beautiful.

However, starting with an angle of M/(73/6) with 2 decimal places means there are only 5 possible values for seconds (s).

For example:

0.00 or X.00 = 60 s, E.G.: 310.00° will result in 60 s.

0.01 or X.Y1 = 36 s, E.G.: 129.61° will result in 36 s.

0.02 or X.Y2 = 12 s, E.G.: 78.92° will result in 12 s.

0.03 or X.Y3 = 48 s, E.G.: 232.43° will result in 48 s.

0.04 or X.Y4 = 24 s, E.G.: 357.74° will result in 24 s.

0.05 or X.Y5 = 60 s, E.G.: 176.85° will result in 60 s.

0.06 or X.Y6 = 36 s, E.G.: 45.16° will result in 36 s.

0.07 or X.Y7 = 12 s, E.G.: 217.97° will result in 12 s.

0.08 or X.Y8 = 48 s, E.G.: 333.28° will result in 48 s.

0.09 or X.Y9 = 24 s, E.G.: 67.89° will result in 24 s.

0.10 or X.Y0 = 60 s, E.G.: 156.40° will result in 60 s.

However, we can start with an angle of M/(73/6) and use 3 decimal places and still have beautiful maths and finish with a whole number for seconds, but the last of the 3 decimal places must always be 5.

For example:

0.00 or X.005 = 60 s, E.G.: 310.005° will result in 18 s.

0.01 or X.Y15 = 36 s, E.G.: 129.615° will result in 54 s.

0.02 or X.Y25 = 12 s, E.G.: 78.925° will result in 30 s.

0.03 or X.Y35 = 48 s, E.G.: 232.435° will result in 6 s.

0.04 or X.Y45 = 24 s, E.G.: 357.745° will result in 42 s.

0.05 or X.Y55 = 60 s, E.G.: 176.855° will result in 18 s.

0.06 or X.Y65 = 36 s, E.G.: 45.165° will result in 54 s.

0.07 or X.Y75 = 12 s, E.G.: 217.975° will result in 30 s.

0.08 or X.Y85 = 48 s, E.G.: 333.285° will result in 6 s.

0.09 or X.Y95 = 24 s, E.G.: 67.895° will result in 42 s.

0.10 or X.Y05 = 60 s, E.G.: 156.405° will result in 18 s.

As you can see having 2 decimals or 3 decimals with the last decimal as 5 goes up alternately by 12 or 6 for seconds. There are 10 values in total for seconds if we are being lazy and starting with an angle of M/(73/6).

For any other whole number for seconds other than the 10 above, we need to start from scratch from the whole number for (s), then divide s/60 then plus k minutes etc. We go up then down or forward and reverse, that is we create the question and then undo it.

The following is the formula for attaining θ in the 4 quadrants of the unit circle:

Q1.

sin^-1(y) = θ

cos^-1(x) = θ

Q2.

sin^-1(y) = π – θ therefore θ = π – sin^-1(y)

cos^-1(x) = θ

Q3.

sin^-1(y) = π – θ therefore θ = π – sin^-1(y)

cos^-1(x) = 2π – θ therefore θ = 2π – cos^-1(x)

Q4.

sin^-1(y) = θ – 2π therefore θ = 2π + sin^-1(y)

cos^-1(x) = 2π – θ therefore θ = 2π – cos^-1(x)

QUESTION 21:

(TRIGONOMETRIC EQUATIONS).

θ = s/60 × 2π

2sin^3(θ) = (-3√3)/4

3cos^3(θ) + 6 = 45/8

a. Solve for θ.

sin^3(θ) = (-3√3)/8

sin(θ) = ³√((-3√3)/8) = -√(3)/2

sin^-1(-√(3)/2) = π – θ = -π/3

θ = π – sin^-1(-√(3)/2) = 4π/3

3cos^3(θ) = 45/8 – 6 = -3/8

cos(θ)^3 = -1/8

cos(θ) = ³√(-1/8) = -1/2

cos^-1(-1/2) = 2π – θ = 2π/3

θ = 2π – cos^-1(-1/2) = 4π/3

b. Solve for s.

4π/3 = s/60 × 2π

2/3 = s/60

s = 2/3 × 60 = 40

c. 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 = 2/3

k = m – s/60 = 43

m = k + s/60 = 131/3

m/60 = 131/180

k = h – m/60 = 15

h = k + m/60 = 2831/180

h/24 = 2831/4320

k = d – h/24 = 17

d = k + h/24 = 76271/4320

d/30 = 76271/129600

k = M – d/30 = 8

M = k + d/30 = 1113071/129600

M/(73/6) = 1113071/1576800

k = y – M/(73/6) = 11380

y = k + M/(73/6) = 17945097071/1576800

d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ.

y = sin(M(73/6) × 2π) = -0.96186457013315

x = cos(M(73/6) × 2π) = -0.27352613901153

Note: to workout nice neat radians fractions instead of ugly decimals you need the Natural Scientific Calculator app! Invite link here:

Natural Scientific Calculator (NSC).

sin^-1(y) = π – θ = -324671π/788400

θ = π – sin^-1(y) = 1113071π/1576800

cos^-1(x) = 2π – θ = 463729π/788400

θ = 2π – cos^-1(x) = 1113071π/1576800

e. Although you have already done it workout M/(73/6), M and k.

M/(73/6) = θ/2π = 1113071/1576800

M = M/(73/6) × (73/6) = 1113071/129600

k = M – d/30 = 8

f. Although you have already done it from (s) upwards, reverse-workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.

d/30 = M – k = 76271/129600

d = (M – k) × 30 = 76271/4320

k = d – h/24 = 17

h/24 = d – k = 2831/4320

h = (d – k) × 24 = 2831/180

k = h – m/60 = 15

m/60 = h – k = 131/180

m = (h – k) × 60 = 131/3

k = m – s/60 = 43

s/60 = m – k = 2/3

s = (m – k) × 60 = 40

t = 11380 years 8 months 17 days 15:43:40

g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?

T^1 = √(A/B) = 10^3 ms

T^-1 = √(B/A) = 10^-3 ks

T^2 = A/B = 10^6 ms²

T^-2 = B/A = 10^-6 ks²

h. Workout the values of t, A and B?

t = A/T^1 = y × 31536000 = 3.58901941420 × 10^11 s

A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms

B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks

i. Check months and days.

y – d/365 = t / 31536000 – d/365 = 11380

d – h/24 = d/365 × 365 – h/24 = 257

h – m/60 = h/24 × 24 – m/60 = 15

m – s/60 = m/60 × 60 – s/60 = 43

s – cs/100 = s/60 × 60 – cs/100 = 40

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 = 358901941420 / 31536000 – d/365 = 11380

d – h/24 = (11,380.7059049974 – 11380) × 365 – h/24 = 257

h – m/60 = (257.655324074074 – 257) × 24 – m/60 = 15

m – s/60 = (15.7277777777777 – 15) × 60 – s/60 = 43

s – cs/100 = (43.6666666666666 – 43) × 60 – cs/100 = 40

t = 11380 years 257 days 15:43:40

Hand written example:

## Author:Technical compassion.

Consider compassion, how can Australopithecus or Lower Palaeolithic man be compassionate toward animals, when they themselves were not yet masters of the animal kingdom or even worse still prey themselves? It is impossible, compassion simply did not exist. Compassion is technical, in that you must, for instance, first attain advanced weapons, technologies and infrastructure such as gunpowder, muskets, rifles, nuclear weapons, automobiles, militaries, police, emergency services, roads, buildings, bridges and skyscrapers etc before you can be compassionate toward animals. It is not a case of hey compassion for compassion’s sake like the Buddha. Compassion is not free of charge, it is a definite and tangible deal or bargain. Only now that I am invincibly safe and secure from wild animals in my city, town or fortress and surrounded by guns, and now that I have an overabundance and surplus of food, energy and resources etc can or will I be compassionate toward animals. It is like saying to ‘bear’ “I have a nuke now, therefore I am compassionate toward you.” This is something bear will never understand, in that it is ironic that once you attain nuclear weapons that you are therefore by definition compassionate toward animals. To reiterate, compassion is something technical, it is only attained through a collective effort, through taming the wild and through civilisation. You can only be compassionate once there is no competition.