Question 21: Trigonometric equations.

QUESTION 21.

TRIGONOMETRY AND RADIANS

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.

QUADRANTS

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:

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