# Questions tagged [trigonometry]

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

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### $\sum_{n=1}^{\infty} \frac{\sin(n^2)}{n^2}$

Question: $$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{n^2}=\,?$$ Previously I calculated a similar summation but it was more luck than wisdom, and insight led me to believe my methods were super incorrect (...
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### What properties can I deduce about an $f(i)$ that satisfies $\sum\limits_{i=1}^{x} \cos(f(i)) = x\cos(\ln(x))$?

I have an $f(i): \mathbb{N} \rightarrow \mathbb{N}$ that satisfies $\sum\limits_{i=1}^{x} \cos(f(i)) = x\cos(\ln(x)), \,\,x \in \mathbb{N}$ What general properties does $f(i)$ satisfy? Can I deduce ...
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### Proving $\lim_{x\rightarrow {\frac{\pi}{2}}^{-}}\tan(x)=+\infty$ and $\lim_{x\rightarrow {-\frac{\pi}{2}}^{+}}\tan(x)=-\infty$ by definition

I have to prove \begin{eqnarray} \lim_{x\rightarrow {\frac{\pi}{2}}^{-}}\tan(x)=+\infty \hspace{1cm} \lim_{x\rightarrow {-\frac{\pi}{2}}^{+}}\tan(x)=-\infty \end{eqnarray} by definition. I don't find ...
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### Solving $c\sin(a(x-b))+d=\frac{c+d}{\frac{\pi}{2a} + b}x$ with $a,b,c,d \in\Bbb{R}$

Help me Solve this Equation $$c\sin(a(x-b))+d=\frac{c+d}{\dfrac{\pi}{2a} + b}x \quad\to\quad x=\dfrac{\pi}{2a} + b$$ $$a,b,c,d \in\Bbb{R}$$ Are you expected to find "closed form" ...
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### How to proof that $\cos x <( \frac{\sin x}{x})^3$ when $0<|x|<\pi\over 2$ [duplicate]

How to proof: $0<|x|<$ $\pi\over 2$ $\quad$ $\Rightarrow \quad$ $\cos x < (\frac{\sin x}{x})^3$
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### How do you simplify expressions like $\tan^{-1}(\cos(x))$ or $\cot^{-1}(\sec^{-1}(x))$? [closed]

In my tutorial, we went over practice problems for inverse trig functions, but I still don't really understand how to arrive at solutions for these types of problems. Is there any methodology/logic I ...
2k views

### Is a triangle with two equal angles always isosceles?

An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles, ...
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### How to solve this spherical trigonometry situation with missing information?

Suppose $b$ and $c$ were given constants. If $C = \theta - y$ (where $\theta$ is given) and $a = 90 \deg - y$, is it possible to solve for $y$? It seems like there must be a solution, since $y$ can't ...
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### How does $\tan^2(x) \sec(x) + \sec^3(x)$ turn in to $2\sec^3(x) - \sec(x)$

Can someone explain how $\tan^2$ disappeared and $\sec^3$ turn into $2\sec^3$ ??? The derivatives of the function $2\sec(x)\tan(x)$ is apparently $2(-\sec(x) + 2\sec^3(x))$
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### Calculating the center of a rotated 2D rectangle given it's bottom left point, angle of rotation and dimensions

I have a rectangle that can be rotated around its center point. I would like to know what the coordinates of the center are, but I'm not sure how to get that with the information available to me. I ...