How to calculate sin(65) without a calculator. I know about the sum and difference formula but I can't think of two values which will be able to use for sin(65). Therefore, I come to the question: How to calculate sin(65) without a calculator.
$$65 = 45 + 20$$

So if you can figure out sine and cosine of 20, you're in good shape. Fortunately,  there are triple-angle formulas:

$$
\sin(3x) = 3\sin x-4\sin^3 x \\
\cos(3x) = \cos^3 x-3\cos x \sin^2 x = \cos^3 x-3\cos x (1 - \cos^2 x)
$$
Since you know $\sin(60) = \sqrt{3}/2$, you know that the sine of 20 -- call it $u$ -- satisfies the equation

$$
\sqrt{3}/2 = 3u-4u^3
$$
From this, you can solve for $u$ (using Cardano's formula for the solution of a cubic). To be honest, this is a complete pain in the neck, but at least it's a route to the solution.
Split it up, into sin 45 plus sin 30. You will get sin 75, then subtract sin 10(you can do by using sin(3x) formula). Then you will get sin 65