# Questions tagged [inverse-trigonometric-functions]

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### Do $\operatorname{arcsec}\left(\frac{2x}{5x+3}\right)$ and $\operatorname{arccos}\left(\frac{5x+3}{2x}\right)$ have the same domain?

Do $$f(x)=\operatorname{arcsec}\left(\frac{2x}{5x+3}\right) \quad\text{and}\quad g(x) = \operatorname{arccos}\left(\frac{5x+3}{2x}\right)$$ have the same domain? I wondered about this while finding ...
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### Behaviour of $\cot^{-1}\cfrac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}$ [duplicate]

$$\operatorname{arccot}\left(\cfrac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right)$$ Why does this function equal to $x/2$ for $x \in \left(0, \pi/4\right)$, and equal ...
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### Is there a closed formula for this sum? [duplicate]

The sum is $$f_{n}=\sum_{k=1}^{n}\arctan\left(\frac{1}{\sqrt{k}}\right)$$ I figured I need a closed formula for this or for the cosine of this whole expression in order to get a polar representation ...
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### Is it the case that $\cos(\arccos(z))=z \iff z\in[-1,1]$? Is my derivation of the falsity of that statement incorrect?

I was recently told that $\cos(\arccos(z))=z \iff z\in[-1,1]$ $\tag1$ I came to a different conclusion by using the following reasoning: Let $z\in\mathbb{C}$ Let $\operatorname{Log}(z)$ denote the ...
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### 2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)

CONTEXT Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
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### How to compute the limit $\lim\limits_{x \to 0^+}{\frac{\cos^{-1} (1-x^2)}{x}}$?

Let $$L=\lim\limits_{x\to 0^+}{\dfrac{\cos^{-1}(1-x^2)}{x}}$$ I tried substituting $x=\sin \theta$ then, $$L=\lim\limits_{\theta \to 0^+}{\dfrac{\cos ^{-1} (\cos^2 \theta)}{\sin \theta}}$$ But I don't ...
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### Faster way to calculate compound function of trigonometric and inverse trigonometric function.

We know that $\arcsin(\sin x) = x$ holds true only for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ for if $x$ does not lie in that interval, we need to add or subtract some multiple of $\pi$, or even need ...
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### The derivative for $\arcsin(2x)$

So by the chain rule I have $$\arcsin'(2x) = \dfrac{1}{\sqrt{1-(2x)^2}} \cdot 2 = \dfrac{2}{\sqrt{1-4x^2}}$$ However, my friend suggested that we can use the following method that arrives at a ...
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### Triangle function that linearly increases in amplitude and decreases in frequency

I am trying to plot a triangle function that linearly increases in amplitude and decreases in frequency. The purpose is to use it in Fusion 360 sketching, but I want to plot it either on my Ti-84 or ...
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### If $\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=2$ then find the value of $(x+y+z)(x-y-z)$
Q): If $\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=2$ then find the value of $(x+y+z)(x-y-z)$. Ans): First of all let me tell everyone that this question is being asked today by the professors of Mathematics of ...