# 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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### $\sin 2x - \tan 2x = -\sin 2x\tan 2x$ trigonometric identity proof

I need to prove $$\sin 2x - \tan 2x = -\sin 2x\tan 2x$$ I tried simplifying $$\sin 2x = 2\sin x\cos x;\quad \tan 2x = \frac{2\tan x}{1-\tan^2x}.$$ But it's so long and complicated that I ...
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### $\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
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### Can you solve a trig equation with a variable both inside a trig function and outside one?

I have the equation: $$d=\frac{t}{2}-\frac{sin(t)}{4}$$ I'm completely failing at how to get this in terms of $t$ I only care about it for values of $0<t<2\pi$ I've seen the graph so I know ...
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### How is the graph of $cot(x)$ valid for negative values of $x$?

In the graph, consider a point between $\frac{-\pi}{2}$ and $-\pi$. We know that $cot(x) = \frac{cos(x)}{sin(x)}$. For negative values of $x$, i.e., $cos(-x)$ is always positive and $sin(-x)$ is ...
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### How do I find angles in the interval, $0^\circ$ $\leq$ $\theta$ $\leq$ $360^\circ$, for $\cot \theta= -\frac{1}{\sqrt3}$ without using a calculator?

If I were to sketch the triangle for this problem on the cartesian plane, it would be in the fourth quadrant (since the side opposite to $\theta$ is negative). From the values of the opposite and ...
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### Evaluate $x^2\left(qy-rz\right)+y^2\left(rz-px\right)+z^2\left(px-qy\right)$, where variables are sines and cosines of $\alpha+2k\pi/3$

Let $x$, $y$, $z$ be the sines of $\alpha$, $\beta$, $\gamma$, and let $p$, $q$, $r$ be the respective cosines, where the angles are in Arithmetic Progression with common difference $\frac{2\pi}{3}$. ...
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### Prove that for all $a > 0$: $\int_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \frac{\pi}{2} - \int_0^a\frac{\sin x}{x}dx$

Prove that for all $a > 0$: $$\int\limits_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \cfrac{\pi}{2} - \int\limits_0^a\cfrac{\sin x}{x}dx$$ I have no idea how to solve it. But the task looks very ...
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### When to substitute with trigo-/hyperbolic-function [closed]

I try to figure out, what indicators could look like to decide if substitution with trigonometrical function or substitution with hyperbolic function works/ works better to integrate a function. Is ...
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### Converting parametric function into cartesian

I am trying to convert the parametric function $x(t) = a\cdot(t - \sin(t)) + b\cdot\cos\left(\frac{t}{2}\right)$ $y(t) = a\cdot\cos(t) + b\cdot\sin\left(\frac{t}{2}\right)$ into a cartesian form. ...
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### Intersection of a line with a Sine function

A sine function is given as $f(x) = 40 + 25.\sin(\pi(x-1.5))$ domain $[0,6]$. I am asked to find a solution for $f(x) < 30$. I know that this is easily solvable with a solver such as in Excel but I ...
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### Taking the limit $\lim_{x \to \infty} x \tan(1/x)$ without L'Hopital's Rule

I have to evaluate the following without L'Hopital's rule $$\lim_{x\to\infty} x\tan(1/x)$$ I can simplify this to be $$\lim_{x\to\infty} x\sin(1/x)$$ because $$\lim_{x\to\infty} \cos(1/x) = 1$$ ...
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### Solution of $1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$

Solve the given inequality: $$1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$$ I am trying to convert L.H.S. in terms of one trigonometric ratio but it is not happening here. Could someone suggest some ...
### Evaluating the indefinite integral $\int\frac{x^2}{\sqrt{4-x^2}}dx$
I think everything I have done is kosher, but unless I am missing an identity it is a different answer than the online quiz and wolfram alpha give. I tried to use the trig substitution  x=2\sin(\...