Q:

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.

A:

$$65 = 45 + 20$$
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. 

A:

$65^\circ$ isn't a nice one, unfortunately: the only constructible angles with natural degree measure are multiples of $3^\circ$.
This is a little complicated to prove directly - a problem since antiquity, the last piece of the puzzle didn't appear until 1837, when Wantzel proved that $20^\circ$ was unconstructible.  Suffice it to say for now that you can get angles of the form $\frac{a}{2^b\cdot3\cdot5\cdot17\cdot257\cdot 65537}$ circles, and $65^\circ$ needs another $3$ on the bottom to get there, being $\frac{13}{72}$ of a circle.
The upshot for you is that you won't be able to "not use a calculator" to find this one.

