How to calculate sin(65) without a calculator.

Question

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.

Answer 1

$$65 = 45 + 20$$

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.

Answer 2

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.)

Answer 3

$65^\circ$ 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.

Answer 4

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