<s>[INST] 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.
 [/INST] Let $i = \sqrt{-1}$. Then from this link, with the methodology explained here:
$$ \sin 65^\circ = -\left(-\frac12 + \frac i2 \sqrt{3}\right) \left( -\frac{1}{32} \sqrt{6} \left(1- \frac{\sqrt 3}{3}\right) + \frac{i}{32} \sqrt{-6 \left( 1 - \frac{\sqrt{3}}{3} \right)^2 + 16 }\right)^{1/3} -\left(-\frac12 - \frac i2 \sqrt{3}\right) \left( -\frac{1}{32} \sqrt{6} \left(1- \frac{\sqrt 3}{3}\right) + \frac{i}{32} \sqrt{-6 \left( 1 - \frac{\sqrt{3}}{3} \right)^2 + 16 }\right)^{1/3} $$
(Note that $65^\circ$ is not constructible, so we should not expect any expression involving a finite amount of additions, multiplications and square roots.)
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