Question

Find the exact value of sin^3 3645° Do I go around the circle 10 times? is there an easier way to do this?

Answer 1

The first thing to note is that, everytime you add one revolution(ie `360` or `2pi`) to an angle you get back to the same angle.

Example: `30°` and `30 + 360° = 390°`
is one and the same angle!
So

Step 1:
Find the number of "`360°s`" there are in that angle, that is; the number of times that angle comes back to the same(duuh).
Do: `3645/360 = 10.125`
This means that you turned round the spot at least `10` times!

Step 2:
Now. After turning around you come back to the original angle: `3645 - 10*360 = 3645 - 3600`
`= 45`

Step 3:
`sin^3(3645) = sin^3(45)`
`= (1/sqrt(2))^3`
`= 1/(2sqrt(2))`

Answer 2

One way to get this is to "go around the circle 10 times" for `3600^@` Then go another `45^@`

Or, understanding that that's what you could do, you could ignore the whole number multiple of `360^@` and just "go to" `45^@`

Or, think less geometrically, use the fact that `sin(A) = sin(A+360^@*n)` for any integer `n`. In this case `n=10`. This amounts to the same thing as "ignoring that `3600^@`

In technical terms: an angle of `3645^@` is co-terminal with an angle of `45^@`. So `sin(3645^@)= sin(45^@)= sqrt2/2`

`sin^3(3645^@)=(sin(3645^@))^3=(sqrt2/2)^3=(2sqrt2)/8=sqrt2/4`