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### Are there any “nonstandard” special angles for which trig functions yield radical expressions?

Everyone learns about the two "special" right triangles at some point in their math education—the $45-45-90$ and $30-60-90$ triangles—for which we can calculate exact trig function outputs. But are ...
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### Easy way of memorizing values of sine, cosine, and tangent

My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday,...
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### An elegant way to solve $\frac {\sqrt3 - 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2$

The question is to find $x\in\left(0,\frac{\pi}{2}\right)$: $$\frac {\sqrt3 - 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2$$ What I did was to take the $\cos x$ fraction to the right ...
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### Remembering /Deriving the values of sine and cosine of 18 degrees,36 degrees,54 degrees,72 degrees

I need to remember the values of sine and cosine of 18 degrees,36 degrees,54 degrees,72 degrees. That is multiples if 18 degrees.Is it possible to derive them in about a minute or so ? Do you use any ...
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### Is there a simple pattern to memorize the sine of $0^\circ$, $15^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$, $90^\circ$?

We know there is a nice pattern to memorize the sine of $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$ as follows. \begin{align} \sin 0^\circ &= \tfrac12\sqrt0\\ \sin 30^\circ &= \...
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### How to find the value of $\sin24^\circ$ [closed]

Is there any method to do it by hand quickly? i want to show the angle $72$ can be trisected by compass and ruler. so i need to find the way to calculate it... help please!
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### Can we express the value of $\sin 1^\circ$ without using the imaginary unit?

I've been playing with sine of integer-degree angles; that is, $\sin\left(\frac{k \pi}{180}\right)$, where $k$ is an integer. I've noticed that you can divide the angle by $2$ and get sine of smaller ...
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### Golden Ratio in trigonometry

Assume that we have been asked to find the value of $\sin (18^\circ)$. We know that there are many ways to find it out. However, I'll be going with golden ratio! Let's draw a triangle whose apical ...
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### Is it just a coincidence or is it related to how values of sine calculated?

Actually, one of my math teacher told me about this. I want to know is there any relationship between this trick and their respective values?
### $\sin(\alpha) = \frac{\sqrt{n}}{k}$, where $n$ and $k$ are integers and $\alpha$ is a rational multiple of $\pi$
It is well known that the solutions of the equation $$\sin\left(\frac\pi x\right)= \frac{\sqrt3}{2}$$ are $$x=\frac{3}{6n+2}, n\in\mathbb{Z}$$ and $$x=\frac{3}{6n+1}, n\in\mathbb{Z}.$$ ...
### Where do the radical expressions for the trig functions of various rational multiples of $\pi$ come from?
So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $\pi$. My ...