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.
[Math] How to calculate sin(65) without a calculator.
trigonometry
Best Answer
$$65 = 45 + 20$$
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.