# Tagged Questions

Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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### Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
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### Question about right triangle and sin(2theta)

This is a pretty basic question but I just wanted clarification. I know that sin(theta) is opposite/hypotenuse regarding right triangles. But what would sin(2theta) be? would it be (opposite/...
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### Why is the phase shift -c/b instead of -c

In a function like $\sin(2x + 3)$ why is the phase shift $\frac{3}{2}$ units to the left instead of 3 units to the left
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### Transform $\tan$ to be continuous between $0$ and $1$

I'm trying to create a $\tan$ function which has asymptotes between $0$ and $1.$ This is the closest I have gotten, but I can see that the asymptote is not actually at $1$ and when $x=0.5,\; y=0.02$. ...
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### Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
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### Trigonometric equation $\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X)$

A trigonometric equation is to be solved, the solution ($X=10^\circ$) is very clear but I need a proper method $$\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X).$$
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### The limit of $(x^2-\tan 2x)/\tan x$ as $x\to0$

I'm stuck in finding the following limit: $$\lim _{x\to 0}\left(\frac{\left(x^2-\tan\left(2x\right)\right)}{\tan\left(x\right)}\right)$$ I am not sure how to do this one help will be appreciated.
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### Proof regarding the function $\cos(1/x)$

Prove that for every number $a>0$ there exists 2 numbers $x,y$ with $0<x,y<a$ for which $f(x)>0$ and $f(y)<0$ with $f = \cos(\dfrac{1}{x})$. How do I go about proving this?
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### Trigonometric eliminations

These are a few problems which I wasn't able to do. I am new to these trigonometric eliminations. I don't really know how to start these problems. I couldn't get pass the first step in some of them.....
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### Find the sum to n terms of the series

Find the sum to n terms of the series $$\frac {\sin x}{\cos x+\cos2x} + \frac {\sin2x}{\cos x+\cos4x} + \frac {\sin3x}{\cos x + \cos6x} +\dotsb$$ How can I solve this? Here is what I did for the ...