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: