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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Why are $\cosh$ and $\cos$ related by the imaginary unit?

I'm interested, why are $\cosh$ and $\cos$ related by the imaginary unit like$$\cos (x)=\cosh (ix)?$$Is there any visual proof? How can be circle related to hyperbola by complex unit?
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Cosine of very large integers, different calculators giving different results

I want to calculate cosine of 452175521116192774 radians (it is around $4.52\cdot10^{17}$) Here is what different calculators say: Wolframalpha Desmos Geogebra Python 3.9 (standard math module) ...
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How to find the closed form of $\int_{-\infty}^{\infty} \frac{x^{2 n+1} \sin x}{\left(1+x^{2}\right)^{n+1}} d x, \textrm{ where }n=0,1,2,3,…?$

In my post, I had found the exact value of the integral $$\displaystyle I:=\int_{-\infty}^{\infty} \frac{x^{3} \sin x}{\left(1+x^{2}\right)^{2}}dx= \frac{\pi}{2 e}\tag*{}$$ by differentiating $J(a)$w....
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Why is the sine of x squared equivalent to sine squared of x?

Why is the sine of x squared equivalent to sine squared of x? How can we verify this or what is the intuition behind it?
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Find n evenly spaced points on circle with radius r

I have a circle that has a given radius $r$. I want to generate $n$ evenly-spaced points along this circle, something like this diagram (with $r=5$ and $n=4$):
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Derivative of $\arcsin(x)=\frac{1}{\cos(x)}$ .

My intuition says that since $\sin(x)$ and $\arcsin(x)$ are inverse of each other, their derivatives must be reciprocal. I have previously proved the fact that $\arcsin(x)'=\frac{1}{\sqrt{1-x^2}}$ but ...
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How to express $(2\cos^2{x}-1)\sin{x}\cos{x}$ as $\frac14 (\sin{4x})$?

I was wondering how I could do this problem. I haven't seen many questions that ask me to express one expression as another in such a format. I know the the two are equal, but I'm not really sure how ...