QUESTION 16.

Note: if you want to create your own trigonometry questions, to save you time, θ or M/(73/6) × 360 should have at least 2 decimal places, otherwise, there are 0 seconds. And 3 decimal places is recommended as the maximum.

Note: radians are very hard, you basically must have the Mathway app to make light work of fractions and radians, also a second scrap piece of paper is recommended.

Question 16:

(TRIGONOMETRY AND RADIANS).

T^1 = √(A/B) = 10^12 ps

k = y – M/(73/6) = 81073

sin(M(73/6) × 2π) = -0.9260024197156

cos(M(73/6) × 2π) = -0.3775175740026

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

b. Workout θ.

c. Workout M/(73/6) and the other decimal remainders and integers (k) of the months, days, hours, minutes and seconds.

d. What is the total value of y?

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

f. Check the days and months.

a.

T^-1 = √(B/A) = 10^-12 Ts

T^2 = A/B = 10^24 ps²

T^-2 = B/A = 10^-24 Ts²

b.

sin^-1(-0.9260024197156) = -3391π/9000

cos^-1(-0.3775175740026) = 5609π/9000

θ = π + 3391π/9000 = 12391π/9000

or

θ = 2π – 5609π/9000 = 12391π/9000

c.

M/(73/6) = (12391π/9000)/2π = 12391/18000

M = M/(73/6) × (73/6) = 904543/108000

k = M – d/30 = 8

d/30 = M – k = 40543/108000

d = (M – k) × 30 = 40543/3000

k = d – h/24 = 11

h/24 = d – k = 943/3600

h = (d – k) × 24 = 943/150

k = h – m/60 = 6

m/60 = h – k = 43/150

m = (h – k) × 60 = 86/5

k = m – s/60 = 17

s/60 = m – k = 1/5

s = (m – k) × 60 = 12

d.

y = k + M/(73/6) = 1459326391/18000

e.

t = A/T^1 = y × 31536000 = 2.556739837032 × 10^12s

A = tT^1 = X × 10^12 × 10^12 = X × 10^24 ps

B = A/T^2 = X × 10^24 / 10^24 = X Ts

f.

y – d/365 = t / 31536000 – d/365 = 81073

d – h/24 = d/365 × 365 – h/24 = 251

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 17

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 = 2556739837032 / 31536000 – d/365 = 81073

d – h/24 = (81073.6883888889 – 81073) × 365 – h/24 = 251

h – m/60 = (251.26194444274 – 251) × 24 – m/60 = 6

m – s/60 = (6.28666662576143 – 6) × 60 – s/60 = 17

s – cs/100 = (17.1999975456856 – 17) × 60 – cs/100 = 12

t = 81073 years 251 days 06:17:12

Hand written example: