QUESTION 19.

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 19:

(CREATE YOUR OWN QUESTIONS).

a. Choose any whole number between 1 and 59 for seconds (s) and divide it 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 = 11/20

m = k + s/60 = 531/20

k = m – s/60 = 26

m/60 = 177/400

h = k + m/60 = 7377/400

k = h – m/60 = 18

h/24 = 2459/3200

d = k + h/24 = 69659/3200

k = d – h/24 = 21

d/30 = 69659/96000

M = k + d/30 = 1029659/96000

k = M – d/30 = 10

M/(73/6) = 1029659/1168000

y = k + M/(73/6) = 67305861659/1168000

k = y – M/(73/6) = 57624

b. Create angle (θ) from M/(73/6) × 2π and get the sine and cosine.

y = sin(M(73/6) × 2π) = -0.67738134317054

x = cos(M(73/6) × 2π) = -0.73563205199642

c. Although you already know it and the object is defeated, workout θ.

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) = θ – 2π = -138341π/584000

θ = 2π + sin^-1(y) = 1029659π/584000

cos^-1(x) = 2π – θ = 138341π/584000

θ = 2π – cos^-1(x) = 1029659π/584000

d. Although you have already done it workout M/(73/6), M and k.

M/(73/6) = θ/2π = 1029659/1168000

M = M/(73/6) × (73/6) = 1029659/96000

k = M – d/30 = 10

e. Although you have already done it from (s), workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.

d/30 = M – k = 69659/96000

d = (M – k) × 30 = 69659/3200

k = d – h/24 = 21

h/24 = d – k = 2459/3200

h = (d – k) × 24 = 7377/400

k = h – m/60 = 18

m/60 = h – k = 177/400

m = (h – k) × 60 = 531/20

k = m – s/60 = 26

s/60 = m – k = 11/20

s = (m – k) × 60 = 33

t = 57624 years 10 months 21 days 18:26:33

e. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?

T^1 = √(A/B) = 10^12 ps

T^-1 = √(B/A) = 10^-12 Ts

T^2 = A/B = 10^24 ps²

T^-2 = B/A = 10^-24 Ts²

g. Workout the values of t, A and B?

t = A/T^1 = y × 31536000 = 1.817258264793 × 10^12 s

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

B = A/T^2 = X × 10^24 / 10^24 = X × 10^0 Ts

h. Check months and days.

y – d/365 = t / 31536000 – d/365 = 57624

d – h/24 = d/365 × 365 – h/24 = 321

h – m/60 = h/24 × 24 – m/60 = 18

m – s/60 = m/60 × 60 – s/60 = 26

s – cs/100 = s/60 × 60 – cs/100 = 33

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 = 1817258264793 / 31536000 – d/365 = 57624

d – h/24 = (57624.8815573630 – 57624) × 365 – h/24 = 321

h – m/60 = (321.7684375 – 321) × 24 – m/60 = 18

m – s/60 = (18.4425 – 18) × 60 – s/60 = 26

s – cs/100 = (26.55 – 26) × 60 – cs/100 = 33

t = 57624 years 321 days 18:26:33

Hand written example: