QUESTION 21.
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 21:
(TRIGONOMETRIC EQUATIONS).
θ = s/60 × 2π
2sin^3(θ) = (-3√3)/4
3cos^3(θ) + 6 = 45/8
a. Solve for θ.
sin^3(θ) = (-3√3)/8
sin(θ) = ³√((-3√3)/8) = -√(3)/2
sin^-1(-√(3)/2) = π – θ = -π/3
θ = π – sin^-1(-√(3)/2) = 4π/3
3cos^3(θ) = 45/8 – 6 = -3/8
cos(θ)^3 = -1/8
cos(θ) = ³√(-1/8) = -1/2
cos^-1(-1/2) = 2π – θ = 2π/3
θ = 2π – cos^-1(-1/2) = 4π/3
b. Solve for s.
4π/3 = s/60 × 2π
2/3 = s/60
s = 2/3 × 60 = 40
c. Re-divide s 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 = 2/3
k = m – s/60 = 43
m = k + s/60 = 131/3
m/60 = 131/180
k = h – m/60 = 15
h = k + m/60 = 2831/180
h/24 = 2831/4320
k = d – h/24 = 17
d = k + h/24 = 76271/4320
d/30 = 76271/129600
k = M – d/30 = 8
M = k + d/30 = 1113071/129600
M/(73/6) = 1113071/1576800
k = y – M/(73/6) = 11380
y = k + M/(73/6) = 17945097071/1576800
d. Create another angle (θ) from M/(73/6) × 2π and get the sine and cosine. Also although you already know it and the object is defeated, workout θ.
y = sin(M(73/6) × 2π) = -0.96186457013315
x = cos(M(73/6) × 2π) = -0.27352613901153
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) = π – θ = -324671π/788400
θ = π – sin^-1(y) = 1113071π/1576800
cos^-1(x) = 2π – θ = 463729π/788400
θ = 2π – cos^-1(x) = 1113071π/1576800
e. Although you have already done it workout M/(73/6), M and k.
M/(73/6) = θ/2π = 1113071/1576800
M = M/(73/6) × (73/6) = 1113071/129600
k = M – d/30 = 8
f. Although you have already done it from (s) upwards, reverse-workout the other decimal remainders and integers (k) of the days, hours, minutes and seconds in reverse order.
d/30 = M – k = 76271/129600
d = (M – k) × 30 = 76271/4320
k = d – h/24 = 17
h/24 = d – k = 2831/4320
h = (d – k) × 24 = 2831/180
k = h – m/60 = 15
m/60 = h – k = 131/180
m = (h – k) × 60 = 131/3
k = m – s/60 = 43
s/60 = m – k = 2/3
s = (m – k) × 60 = 40
t = 11380 years 8 months 17 days 15:43:40
g. Choose and workout the magnitudes of T^1, T^-1, T^2 and T^-2?
T^1 = √(A/B) = 10^3 ms
T^-1 = √(B/A) = 10^-3 ks
T^2 = A/B = 10^6 ms²
T^-2 = B/A = 10^-6 ks²
h. Workout the values of t, A and B?
t = A/T^1 = y × 31536000 = 3.58901941420 × 10^11 s
A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms
B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks
i. Check months and days.
y – d/365 = t / 31536000 – d/365 = 11380
d – h/24 = d/365 × 365 – h/24 = 257
h – m/60 = h/24 × 24 – m/60 = 15
m – s/60 = m/60 × 60 – s/60 = 43
s – cs/100 = s/60 × 60 – cs/100 = 40
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 = 358901941420 / 31536000 – d/365 = 11380
d – h/24 = (11,380.7059049974 – 11380) × 365 – h/24 = 257
h – m/60 = (257.655324074074 – 257) × 24 – m/60 = 15
m – s/60 = (15.7277777777777 – 15) × 60 – s/60 = 43
s – cs/100 = (43.6666666666666 – 43) × 60 – cs/100 = 40
t = 11380 years 257 days 15:43:40
Hand written example:
