QUESTION 17.

TRIGONOMETRY AND RADIANS

To create your own trigonometry questions and to save you time, know that θ or (M/(73/6) × 360) should have 2 decimal places. Any less and there are 0 seconds, anymore and the maths is not beautiful.

Note: 2 decimal places this 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.

Note: the Mathway app makes light work of fractions and radians, also a second scrap piece of paper is recommended.

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 17:

(TRIGONOMETRY AND RADIANS).

T^1 = √(A/B) = 10^21 zs

k = y – M/(73/6) = 80723

sin(M(73/6) × 2π) = -0.602929541689

cos(M(73/6) × 2π) = -0.79779443953857

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(-0.602929541689) = θ – 2π = -103π/500

θ = 2π – 103π/500 = 897π/500

cos^-1(-0.79779443953857) = 2π – θ = 103π/500

θ = 2π – 103π/500 = 897π/500

b.

M/(73/6) = θ/2π = (897π/500)/2π = 897/1000

M = M/(73/6) × (73/6) = 21827/2000

k = M – d/30 = 10

c.

d/30 = M – k = 1827/2000

d = (M – k) × 30 = 5481/200

k = d – h/24 = 27

h/24 = d – k = 81/200

h = (d – k) × 24 = 243/25

k = h – m/60 = 9

m/60 = h – k = 18/25

m = (h – k) × 60 = 216/5

k = m – s/60 = 43

s/60 = m – k = 1/5

s = (m – k) × 60 = 12

t = 80723 years 10 months 27 days 09:43:12

d.

y = k + M/(73/6) = 80723897/1000

e.

T^-1 = √(B/A) = 10^-21 Zs

T^2 = A/B = 10^42 zs²

T^-2 = B/A = 10^-42 Zs²

f.

t = A/T^1 = y × 31536000 = 2.545708815792 × 10^12 s

A = tT^1 = X × 10^12 × 10^21 = X × 10^33 zs

B = A/T^2 = X × 10^33 / 10^42 = X × 10^-9 Zs

g.

y – d/365 = t / 31536000 – d/365 = 80723

d – h/24 = d/365 × 365 – h/24 = 327

h – m/60 = h/24 × 24 – m/60 = 9

m – s/60 = m/60 × 60 – s/60 = 43

s – cs/100 = s/60 × 60 – cs/100 = 12

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 = 2545708815792 / 31536000 – d/365 = 80723

d – h/24 = (80723.897- 80723) × 365 – h/24 = 327

h – m/60 = (327.40499999898 – 327) × 24 – m/60 = 9

m – s/60 = (9.71999997552484 – 9) × 60 – s/60 = 43

s – cs/100 = (43.1999985314906 – 43) × 60 – cs/100 = 12

t = 80723 years 327 days 09:43:12

Hand written example: