Question

What is the exact value of `sin 65^@` ?

Answer 1

There is no exact formula for `sin 65^@` without using trigonometric or related functions.

Explanation:

Unfortunately, `65^@` is an angle for which you cannot find the exact value of `sin theta`.

For angles which are a whole number of degrees, exact formulae can be found if and only if the angle is a multiple of `3^@`.

Note that we have exact values:

`sin 45^@ = cos 45^@ = sqrt(2)/2`

`sin 30^@ = 1/2`

`cos 30^@ = sqrt(3)/2`

Hence we can use the difference formulae to find:

`sin 15^@ = 1/4(sqrt(6)-sqrt(2))`

`cos 15^@ = 1/4(sqrt(6)+sqrt(2))`

In addition, by looking at the regular pentagon, we can find:

`sin 18^@ = 1/4(sqrt(5)-1)`

`cos 18^@ = 1/4 sqrt(10+2sqrt(5))`

See https://socratic.org/s/aFZNCmPD for details.

I won't multiply it out, but then we can use the difference formulae again to find exact formulae:

`sin 3^@ = sin 18^@ cos 15^@ - cos 18^@ sin 15^@`

`cos 3^@ = cos 18^@ cos 15^@ + sin 18^@ sin 15^@`

From these values you can find trigonometric values for any multiple of `3^@`.

To get values for `sin 1^@`, `cos 1^@` or any other whole number of degrees that is not a multiple of `3` involves taking the cube root of a complex number. You can certainly express it in that form, but it tells you nothing substantial: To find the formula in terms of real numbers will involve trigonometric functions again.