QUESTION 18.

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 18:

(TRIGONOMETRY AND RADIANS).

T^1 = √(A/B) = 10^18 as

k = y – M/(73/6) = 57892

sin(M(73/6) × 2π) = -0.5278455119451

cos(M(73/6) × 2π) = -0.8493404002633

a. Workout θ.

b. Workout M/(73/6), M and k.

c. Workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds.

d. What is the total value of y?

e. What are the magnitudes of T^-1, T^2 and T^-2?

f. What are the values of t, A and B?

g. Check the days and months.

Note: to figure out a fraction radian from a decimal radian, divide the decimal radian by 2π then convert the resulting decimal into a fraction.

For example:

0.6471680866395 / 2π = 0.103 = 103/1000

Then multiply the fraction by 2π.

radian = 103π/500

a.

sin^-1(-0.5278455119451) = π – θ = -177π/1000

θ = π – (-177π/1000) = 1177π/1000

cos^-1(-0.8493404002633) = 2π – θ = 823π/1000

θ = 2π – 823π/1000 = 1177π/1000

b.

M/(73/6) = θ/2π = (1177π/1000)/2π = 1177/2000

M = M/(73/6) × (73/6) = 85921/12000

k = M – d/30 = 7

c.

d/30 = M – k = 1921/12000

d = (M – k) × 30 = 1921/400

k = d – h/24 = 4

h/24 = d – k = 321/400

h = (d – k) × 24 = 963/50

k = h – m/60 = 19

m/60 = h – k = 13/50

m = (h – k) × 60 = 78/5

k = m – s/60 = 15

s/60 = m – k = 3/5

s = (m – k) × 60 = 36

t = 57892 years 7 months 4 days 19:15:36

d.

y = k + M/(73/6) = 115785177/2000

e.

T^-1 = √(B/A) = 10^-18 Es

T^2 = A/B = 10^36 as²

T^-2 = B/A = 10^-36 Es²

f.

t = A/T^1 = y × 31536000 = 1.825700670936 × 10^12 s

A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as

B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es

g.

y – d/365 = t / 31536000 – d/365 = 57892

d – h/24 = d/365 × 365 – h/24 = 214

h – m/60 = h/24 × 24 – m/60 = 19

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 36

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 = 1825700670936 / 31536000 – d/365 = 57892

d – h/24 = (57892.5885 – 57892) × 365 – h/24 = 214

h – m/60 = (214.802499999278 – 214) × 24 – m/60 = 19

m – s/60 = (19.2599999826634 – 19) × 60 – s/60 = 15

s – cs/100 = (15.5999989598058 – 15) × 60 – cs/100 = 36

t = 57892 years 214 days 19:15:36

Hand written example: