Linked Questions

40
votes
11answers
8k views

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 ...
45
votes
12answers
305k views

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,...
5
votes
3answers
2k views

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 ...
1
vote
4answers
857 views

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 ...
5
votes
2answers
113 views

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 &= \...
0
votes
1answer
275 views

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!
2
votes
1answer
141 views

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 ...
0
votes
0answers
333 views

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 ...
3
votes
1answer
81 views

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?
1
vote
1answer
52 views

$\sin(\alpha) = \frac{\sqrt{n}}{k}$, 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 $$ x=\frac{3}{6n+1}, n\in\mathbb{Z}. $$ ...
0
votes
0answers
49 views

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 ...