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:

Author:

The law in one frame of reference or time period is not the law in another frame of reference or time period. The law changes over space and time. The law is not absolute. The law is relative. The law is flexible.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s