Questions

Question 18: Trigonometry and radians.

QUESTION 18.

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 18:

(TRIGONOMETRY AND RADIANS).

T^1 = √(A/B) = 10^18 as

k = y – M/(73/6) = 57892

sin(M(73/6) × 2π) = -0.5278455119451

cos(M(73/6) × 2π) = -0.8493404002633

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(-0.5278455119451) = π – θ = -177π/1000

θ = π – (-177π/1000) = 1177π/1000

cos^-1(-0.8493404002633) = 2π – θ = 823π/1000

θ = 2π – 823π/1000 = 1177π/1000

b.

M/(73/6) = θ/2π = (1177π/1000)/2π = 1177/2000

M = M/(73/6) × (73/6) = 85921/12000

k = M – d/30 = 7

c.

d/30 = M – k = 1921/12000

d = (M – k) × 30 = 1921/400

k = d – h/24 = 4

h/24 = d – k = 321/400

h = (d – k) × 24 = 963/50

k = h – m/60 = 19

m/60 = h – k = 13/50

m = (h – k) × 60 = 78/5

k = m – s/60 = 15

s/60 = m – k = 3/5

s = (m – k) × 60 = 36

t = 57892 years 7 months 4 days 19:15:36

d.

y = k + M/(73/6) = 115785177/2000

e.

T^-1 = √(B/A) = 10^-18 Es

T^2 = A/B = 10^36 as²

T^-2 = B/A = 10^-36 Es²

f.

t = A/T^1 = y × 31536000 = 1.825700670936 × 10^12 s

A = tT^1 = X × 10^12 × 10^18 = X × 10^30 as

B = A/T^2 = X × 10^30 / 10^36 = X × 10^-6 Es

g.

y – d/365 = t / 31536000 – d/365 = 57892

d – h/24 = d/365 × 365 – h/24 = 214

h – m/60 = h/24 × 24 – m/60 = 19

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 36

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 = 1825700670936 / 31536000 – d/365 = 57892

d – h/24 = (57892.5885 – 57892) × 365 – h/24 = 214

h – m/60 = (214.802499999278 – 214) × 24 – m/60 = 19

m – s/60 = (19.2599999826634 – 19) × 60 – s/60 = 15

s – cs/100 = (15.5999989598058 – 15) × 60 – cs/100 = 36

t = 57892 years 214 days 19:15:36

Hand written example:

6B5F109F-83BF-44D9-BF59-0E911A2CD219

Question 17: Trigonometry and radians.

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:

2D5AE5BA-753E-46DA-AD81-D943B981AB39

Question 16: Trigonometry and radians.

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:

A4277A34-1D3B-4A6D-9A71-905726079058

Question 15: Trigonometry.

QUESTION 15.

(TRIGONOMETRIC).

T^1 = √(A/B) = 10^15 fs

k = y – M/(73/6) = 4623

sin(M(73/6) × 360) = -0.3050305188087

cos(M(73/6) × 360) = 0.95234257627983

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 answer.

a.

T^-1 = √(B/A) = 10^-15 Ps

T^2 = A/B = 10^30 fs²

T^-2 = B/A = 10^-30 Ps²

b.

sin^-1(-0.3050305188087) = -17.76°

cos^-1(0.95234257627983) = 17.76°

θ = 360 – 17.76 = 342.24° = 8556/25°

c.

M/(73/6) = (8556/25)/360 = 713/750

M = M/(73/6) × (73/6) = 52049/4500

k = M – d/30 = 11

d/30 = M – k = 2549/4500

d = (M – k) × 30 = 2549/150

k = d – h/24 = 16

h/24 = d – k = 149/150

h = (d – k) × 24 = 596/24

k = h – m/60 = 23

m/60 = h – k = 21/25

m = (h – k) × 60 = 252/5

k = m – s/60 = 50

s/60 = m – k = 2/5

s = (m – k) × 60 = 24

d.

y = k + M/(73/6) = 3467963/750

e.

t = A/T^1 = y × 31536000 = 1.45820908224 × 10^11s

A = tT^1 = X × 10^11 × 10^15 = X × 10^26 fs

B = A/T^2 = X × 10^26 / 10^30 = X × 10^-4 Ps

f.

y – M/(73/6) = t / 31536000 – M/(73/6) = 4623

M – d/30 = M/(73/6) × (73/6) – d/30 = 11

d – h/24 = d/30 × 30 – h/24 = 16

h – m/60 = h/24 × 24 – m/60 = 23

m – s/60 = m/60 × 60 – s/60 = 50

s – cs/100 = s/60 × 60 – cs/100 = 24

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 – M/(73/6) = 145820908224 / 31536000 – M/(73/6) = 4623

M – d/30 = (4623.95066666667 – 4623) × (73/6) – d/30 = 11

d – h/24 = (11.5664444444439 – 11) × 30 – h/24 = 16

h – m/60 = (16.9933333333165 – 16) × 24 – m/60 = 23

m – s/60 = (23.8399999995959 – 23) × 60 – s/60 = 50

s – cs/100 = (50.3999999757542 – 50) × 60 – cs/100 = 24

t = 4623 years 11 months 16 days 23:50:24

Hand written example:

5DFD87F4-FDBB-4AA8-A32C-20CC386893C7

Question 14: Trigonometry.

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:

F356C911-C3BA-4ECC-9FCF-EC84203A3C1E

Question 13: Trigonometry reverse.

QUESTION 13.

(Trigonometry reverse).

T^1 = √(A/B) = 10^9 ns

k = y – M/(73/6) = 4248

sin(θ) = 0.46719680793178

cos(θ) = -0.8841533479314

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

b. Workout θ and M/(73/6).

c. What is the total value of y?

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

e. Workout the decimal remainders and integers (k) of the months, days, hours, minutes and seconds.

f. Check if remainder days and months match.

a.

T^-1 = √(B/A) = 10^-9 Gs

T^2 = A/B = 10^18 ns²

T^-2 = B/A = 10^-18 Gs²

b.

sin^-1(0.46719680793178) = 27.852488584475°

cos^-1(-0.8841533479314) = 152.147511415521° = 6664061/43800°

θ = 152.147511415521° = 6664061/43800°

M/(73/6) = (6664061/43800)/360 = 6664061/15768000

c.

y = k + M/(73/6) = 66989128061/15768000

d.

t = A/T^1 = y × 31536000 = 1.33978256122 × 10^11s

A = tT^1 = X × 10^11 × 10^9 = X × 10^20 ns

B = A/T^2 = X × 10^20 / 10^18 = X × 10^2 Gs

e.

M = M/(73/6) × (73/6) = 6664061/1296000

k = M – d/30 = 5

d/30 = M – k = 184061/1296000

d = (M – k) × 30 = 184061/43200

k = d – h/24 = 4

h/24 = d – k = 11261/43200

h = (d – k) × 24 = 11261/1800

k = h – m/60 = 6

m/60 = h – k = 461/1800

m = (h – k) × 60 = 461/30

k = m – s/60 = 15

s/60 = m – k = 11/30

s = (m – k) × 60 = 22

f.

y – d/365 = t / 31536000 – d/365 = 4248

d – h/24 = d/365 × 365 – h/24 = 154

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

For example:

d = m × 30 + d

d = 5 × 30 + 4 = 154

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 = 133978256122 / 31536000 – d/365 = 4248

d – h/24 = (4248.42263197615 – 4248) × 365 – h/24 = 154

h – m/60 = (154.260671296452 – 154) × 24 – m/60 = 6

m – s/60 = (6.25611111483886 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903315 – 15) × 60 – cs/100 = 22

t = 4248 years 154 days 06:15:22

Hand written example:

A2B5ABFC-3290-4248-B1E9-4156A9F94181

Question 12: Dates and times trigonometry.

QUESTION 12.

(DATES AND TIMES TRIGONOMETRY).

11/2/1544 09:16:32 – 15/7/5792 15:31:54

T^1 = √(A/B) = 10^3 ms

a. What are the integers (k) of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What are the values of s, m, h, d, M and y and what are the angles of their remainders or decimals in degrees and radians? Give sin(θ) and cos(θ).

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

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

e. Reverse or undo t back into y/M/d/h/m/s integers (k) format.

f. Check that the remainder months add up to the remainder days.

Answer:

a.

k = y – M/(73/6) = 5792 – 1544 = 4248

k = M – d/30 = 7 – 2 = 5

k = d – h/24 = 15 – 11 = 4

k = h – m/60 = 15 – 9 = 6

k = m – s/60 = 31 – 16 = 15

k = s – cs/100 = 54 – 32 = 22

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 30 days in a month, therefore, we are using conversion factor of 73/6 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/31.

Note: to convert a remainder decimal of a unit time into an angle of degrees and radians we times the decimal by 360 and 2π respectively.

For example:

s/60 = 11/30 × 360 = 132° Or s/60 = 11/30 × 2π = 11π/15

b.

s/60 = 22/60 = 11/30

θ = 132° or 11π/15

sin(θ) = 0.74314482547739

cos(θ) = -0.6691306063589

m = k + s/60 = 461/30

m/60 = (461/30)/60 = 461/1800

θ = 461/5° or 922π/5

sin(θ) = 0.99926291641062

cos(θ) = -0.0383878090875

h = k + m/60 = 11261/1800

h/24 = (11261/1800)/24 = 11261/43200

θ = 11261/120° or 11261π/21600

sin(θ) = 0.99775300871239

cos(θ) = -0.0669995045159

d = k + h/24 = 184061/43200

d/30 = (184061/43200)/30 = 184061/1296000

θ = 184061/3600° or 184061π/648000

sin(θ) = 0.77855054474696

cos(θ) = 0.62758190642674

M = k + d/30 = 6664061/1296000

M/(73/6) = (6664061/1296000)/(73/6) = 6664061/15768000

θ = 6664061/43800° or 6664061π/7884000

sin(θ) = 0.46719680793178

cos(θ) = -0.8841533479314

y = k + M/(73/6) = 66989128061/15768000

c.

T^-1 = √(B/A) = 10^-3 ks

T^2 = A/B = 10^6 ms²

T^-2 = B/A = 10^-6 ks²

d.

t = A/T^1 = y × 31536000 = 1.33978256122 × 10^11 s

X = 1.33978256122

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

e.

y – M/(73/6) = t / 31536000 – M/(73/6) = 4248

M – d/30 = M/(73/6) × (73/6) – d/30 = 5

d – h/24 = d/30 × 30 – h/24 = 4

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

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 – M/(73/6) = 133978256122 / 31536000 – M/(73/6) = 4248

M – d/30 = (4248.42263197615 – 4248) × (73/6) – d/30 = 5

d – h/24 = (5.14202237654839 – 5) × 30 – h/24 = 4

h – m/60 = (4.26067129645161 – 4) × 24 – m/60 = 6

m – s/60 = (6.25611111483865 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903188 – 15) × 60 – cs/100 = 22

Then insert the integers (k) into the y/M/d/h/m/s format:

t = 4248 years 5 months 4 days 06:15:22

f.

y – d/365 = t / 31536000×- d/365 = 4248

d – h/24 = d/365 × 365 – h/24 = 154

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

For example:

d = m × 30 + d

d = 5 × 30 + 4 = 154

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 = 133978256122 / 31536000 – d/365 = 4248

d – h/24 = (4248.42263197615 – 4248) × 365 – h/24 = 154

h – m/60 = (154.260671296452 – 154) × 24 – m/60 = 6

m – s/60 = (6.25611111483886 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903315 – 15) × 60 – cs/100 = 22

t = 4248 years 154 days 06:15:22

Hand written example:

5A13AA24-3926-4FF8-8BF0-65135CA0118A