In my class we are not allowed to use calculators on math tests. So I was hoping to find a way to calculate the sin or cos of an angle without using a calculator. I would also prefer not to use memorize a unit circle. I've learned the rules for the special triangles but how do I figure out problems like sin(15) or cos(65) etc.?

Thank you for the opportunity to help you with your question!

Use the special angles. These are 0, 30, 45, 60, 90

and use the angle difference identities like Sin(A-B) = Sin A cos B - Cos A Sin B There are more formulas for Sin (A+B), cos (A+B), etc you must have been taught these.

.
sin 15 = sin (45-30) = sin 45 cos 30 - sin 30 cos 45
simplify these angles ((sqrt 2)/2 * (sqrt 3)/2 - 1/2 *(sqrt 2)/2)
Simplify these and you will have your answer. Combine like terms.

Let me know incase you need any further help ! Thanks :)