QUESTION 14.

(TRIGONOMETRIC).

T^1 = √(A/B) = 10^6 μs

k = y – M/(73/6) = 3972

sin(θ) = -0.901681645086

cos(θ) = 0.43240052140917

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

b. Workout θ.

c. Workout the 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 if remainder days and months match.

a.

T^-1 = √(B/A) = 10^-6 Ms

T^2 = A/B = 10^12 μs²

T^-2 = B/A = 10^-12 Ms²

b.

sin^-1(-0.901681645086) = -64.38°

cos^-1(0.43240052140917) = 64.38°

θ = 360 – 64.38 = 295.62° = 14781/50°

c.

M/(73/6) = (14781/50)/360 = 4927/6000

M = M/(73/6) × (73/6) = 359671/36000

k = M – d/30 = 9

d/30 = M – k = 35671/36000

d = (M – k) × 30 = 35671/1200

k = d – h/24 = 29

h/24 = d – k = 871/1200

h = (d – k) × 24 = 871/50

k = h – m/60 = 17

m/60 = h – k = 21/50

m = (h – k) × 60 = 126/5

k = m – s/60 = 25

s/60 = m – k = 1/5

s = (m – k) × 60 = 12

d.

y = k + M/(73/6) = 23836927/6000

e.

t = A/T^1 = y × 31536000 = 1.25286888312 × 10^11s

A = tT^1 = X × 10^11 × 10^6 = X × 10^17 μs

B = A/T^2 = X × 10^17 / 10^12 = X × 10^5 Ms

f.

y – d/365 = t / 31536000×- d/365 = 3972

d – h/24 = d/365 × 365 – h/24 = 299

h – m/60 = h/24 × 24 – m/60 = 17

m – s/60 = m/60 × 60 – s/60 = 25

s – cs/100 = s/60 × 60 – cs/100 = 12

For example:

d = m × 30 + d

d = 9 × 30 + 29 = 299

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 = 125286888312 / 31536000 – d/365 = 3972

d – h/24 = (3972.82116666667 – 3972) × 365 – h/24 = 299

h – m/60 = (299.725833333314 – 299) × 24 – m/60 = 17

m – s/60 = (17.4199999995326 – 17) × 60 – s/60 = 25

s – cs/100 = (25.1999999719555 – 25) × 60 – cs/100 = 12

t = 3972 years 299 days 17:25:12

Hand written example: