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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 ...
improvement dude's user avatar
-1 votes
1 answer
88 views

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 ...
Aryan Agarwal's user avatar
2 votes
2 answers
179 views

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 ...
עמית חי לרמן's user avatar
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1 answer
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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 ...
Simon M's user avatar
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Derivative of $y= \frac{x}{1+x^2} - \arctan x$

Let $y= \frac{x}{1+x^2} - \arctan x$ Differentiating using quotient rule on the first term, $$\begin{align}\frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{(1+x^2)(1)-x(2x)}{(1+x^2)^2}-\frac{1}{1+x^2}\\&...
Haider's user avatar
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Evaluate: $\int{\cos{\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right)}dx}$

Evaluate: $$\int{\cos{\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right)}dx}.$$ Method 1 Put $x=\cos(2\theta)$; then $\mathrm dx=-2\sin(2\theta)\,\mathrm d\theta$ $$\begin{align*} I&= \int\cos(2\cot^{-...
Daksh's user avatar
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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 $(-1)^k$ ...
olivierlambert's user avatar
3 votes
2 answers
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Integrating $\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$ two ways gives very different-looking answers. How to show algebraically they differ by a constant?

From Apostol's Calculus Volume 1 2nd ed., 6.22 #46, the task is to integrate $$\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$$ Method 1: The provided hint is to use the substitution $x-a=(b-a)\sin^2(u)$. Thus $b-...
newmacuser's user avatar
-1 votes
3 answers
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Find the value of $\int_{-\infty}^{\infty} \cos^{-1}\left(\cos\left(\frac{24+4 x^2}{4+x^2}\right)\right)dx$. [closed]

Find the value of $\int_{-\infty}^{\infty} \cos^{-1}\left(\cos\left(\frac{24+4 x^2}{4+x^2}\right)\right)dx$. I tried separating the limits or converting it to a different trigonometric function. ...
bhargavi narayanan's user avatar
7 votes
1 answer
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Solving inverse trigonometric equation involving arccotangent.

$\tan^{-1}(x+1)+\cot^{-1}(\frac{1}{x-1})=\tan^{-1}(\frac{8}{31})$ One thing to clear is that the range of arccotangent is $(0,\pi)$. I am so sad because I can't use Wolfram to cross check my answers. ...
Darshit Sharma's user avatar
17 votes
2 answers
1k views

Crazy integral with nested radicals and inverse sines

Recently a friend who is writing a book on integrals added this problem to his book: $$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{x}}}\ dx=\frac{3\pi}{16}$$ After a while, when trying to generalize, I was able ...
pvr95's user avatar
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3 answers
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Arc cosine of the negative cosine

I'm trying to simplify the following equation $$\frac{1}{2\pi}(2\cos^{-1}(-\cos x) - \sin(2\cos^{-1}(-\cos x)))$$ with $0 \le x \le \pi$. The negative sign in $\cos^{-1}(-\cos x)$ is what is ...
rdemo's user avatar
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Is a function differentiable at a if we can't define either one of $\lim\limits_{x \to a^+}$ or $\lim\limits_{x \to a^-}$

This question came in my test and I applied what I was told by my teacher. The question is If $f (x) = |1 – x|$, then the number of points where $g (x) = \arcsin(f(|x|))$ is non-differentiable, are ...
satyam singh's user avatar
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Simplify $\arctan(x + b)$ where $b$ is a constant positive integer?

I was wondering if it was possible to simplify $\arctan(x + b)$ so that I can factor out the $x$ from the expression entirely. I tried searching a rule for the $\arctan$ of a sum but came up empty-...
Arjun Krishnan's user avatar
4 votes
2 answers
99 views

Solving an exotic trigonometric function or a sextic

I'm an aerospace engineering student and I've been doing some analytical work on an interdisciplinary problem involving orbital mechanics and electromagnetism. In the final part of my work, I ended up ...
Eric D'Antona's user avatar
1 vote
1 answer
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atan2 function to find 2 circles centers from 2 points that are less than $2r$ apart

Given 2 points $p$, $q$, and $r$ that are less than $2r$ apart, where $r$ is the radius of a circle. I am asked to find 2 circles centers $c_1$ and $c_2$ that are where the first center is less than $...
Avv's user avatar
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-1 votes
3 answers
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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 ...
Jesko's user avatar
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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 ...
Ansh's user avatar
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3 answers
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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 ...
xxdnyudwuw's user avatar
0 votes
1 answer
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What is the different between these two forms of the derivative of arcsine?

It is known that the derivative of an inverse function is given as $$ g'(y)=\frac{1}{f'(x)} \implies \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} $$ So if $\arcsin(y)$ is differentiated: $$ \arcsin(y)' = \...
thewhale's user avatar
9 votes
4 answers
1k views

How to prove the two answers to an integral are equivalent

I'm trying to do the integral: $$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$ So I try two ways to do it, the first method I used is to multiply $e^x$ on both sides first. $$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$...
ACgroup's user avatar
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3 answers
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Why are these primitives containing $\arcsin x$ equal up to a constant?

While trying to solve $\displaystyle\int\sqrt{14x-x^2}\;dx$, I obtained three different primitives in three different ways: Method 1: completing the square $c(x)=\dfrac{1}{2}\left[49\arcsin\left(\...
orion2112's user avatar
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1 answer
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ODE with trigonometric functions

I'm solving the following ODE: $$(1 + y^2) = (\arctan(y) - x)\frac{dy}{dx},$$ with $y(x)$ being the unknown function. The Solutions sheet recommends substituting $u(x) = \arctan(y(x))$, so that $\frac{...
Riccardo Iorio's user avatar
3 votes
1 answer
98 views

Why do we turn $|\tan^{-1}\theta|$ to $-\tan^{-1}θ$?

In this question, why did we turn $|\tan^{-1}\theta|$ to $-\tan^{-1}θ$ in the highlighted step? Why is there a negative sign? I first thought that the negative sign was because $\frac\pi2 < \theta ...
jaik_2000's user avatar
2 votes
2 answers
214 views

$\frac{\sin3x + 2}{\sin x+2} = 2023^{\sin x-\sin3x}$

The statement of the problem : Solve in $\mathbb R$ the following equation : $$\frac{\sin3x + 2}{\sin x+2} = 2023^{\sin x-\sin3x}$$ My approach : To simplify, we will use the fact that $\sin3x = 3*\...
Last X's user avatar
  • 311
1 vote
2 answers
39 views

Composition of Taylor expansions of trigonometric functions and their inverses.

I was trying to check whether $ arc sin (sin \theta) = \theta$ and $arc cos(cos \theta) = \theta $ is satisfied when I compose Taylor series expansion of these functions, i.e.: For sine: $$ x = sin \...
Mateusz Wyszyński's user avatar
0 votes
2 answers
76 views

Creating a relevant right triangle to evaluate $\sec\left(\arctan\frac{4}{3}\right)$

I am trying to solve the following: $$\sec\left(\arctan\left(\frac{4}{3}\right)\right)$$ The problem tells me to use a relevant right triangle, but I am curious as to if I need to create a right ...
MSM's user avatar
  • 25
6 votes
5 answers
251 views

Solve the equation $\arcsin\bigg(\dfrac{x+1}{\sqrt{x^2+2x+2}}\bigg)-\arcsin\bigg(\dfrac{x}{\sqrt{x^2+1}}\bigg)=\dfrac{\pi}{4}$

Solve the equation $$\arcsin\bigg(\dfrac{x+1}{\sqrt{x^2+2x+2}}\bigg)-\arcsin\bigg(\dfrac{x}{\sqrt{x^2+1}}\bigg)=\dfrac{\pi}{4}$$ My solution: I converted this equation in terms of $\arctan$ and ...
mathophile's user avatar
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1 vote
1 answer
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How to find when does the derivatives of functions involving some inverse trignometric functions be negative or positive?

Consider y = cos-1(sin x): I got dy/dx = +/- 1, but my textbook only mentions -1 as a solution. Here's how they arrived at it: y = cos-1(sin x) = cos-1(cos(π/2-x)) = π/2 - x => dy/dx = -1 I got +/- ...
Cinverse's user avatar
  • 181
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0 answers
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How to deal with error caused by sign change in arcsin?

The problem happens in antenna array signal processing. I have two antennas A and B. According to the complex signal received, I have the phases pA and pB. Both pA and pB are between -$\pi$ and $\pi$. ...
Beck Chen's user avatar
0 votes
1 answer
38 views

Arcsine expressed by complex numbers.

I have found this expression (I am worried about domain issues or any algebraic mistake I make)by playing around with the formula for sine, i.e- $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$ From this, I got-&...
Fusion crafter's user avatar
1 vote
1 answer
92 views

Using chain rule to find $\frac{dy}{dx}$ of an inverse sine, got negative of the actual solution

Problem Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$, where $y = \sin^{-1}\biggl(\dfrac{2x}{1+x^2}\biggr)$ Given solution $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2}{1+x^2}$ My approach I got the ...
Cinverse's user avatar
  • 181
0 votes
2 answers
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How to find all solutions to $\cos(x) = a$?

I have the following question at hand: Find all extrema of the function $f(x) = x - 2\sin(x+ \frac{\pi}{4})$. That amounts to solving $\cos(x + \frac{\pi}{4}) = \frac{1}{2}$. But simply using $\arccos(...
T. Feix's user avatar
  • 121
1 vote
1 answer
78 views

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 ...
J. Tanin's user avatar
0 votes
2 answers
87 views

Is it possible to cancel out the nested $\arccos$ inside the $\cos$ and $\sin$ in this expression?

I have the expression $\sqrt{\frac{23 + 16 × \sqrt{2}}{68}} × \sin\left(a × \arccos\left(\frac{\sqrt{8} - 1}{4}\right)\right) - \frac{1}{2} × \cos\left(a × \arccos\left(\frac{\sqrt{8} - 1}{4}\right)\...
Lawton's user avatar
  • 1,861
1 vote
3 answers
238 views

Trying to find an identity for $\arctan(ab)$ in terms of any other trigonometric function

I am trying to find an identity for $\arctan(ab)$ in terms of any trigonometric function involving $a$ and $b$ being separate. I have attempted doing so using $(\arctan(a)+\arctan(b))^2$ but that got ...
Yajat Shamji's user avatar
2 votes
0 answers
79 views

Find the range of the function $f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$

Find range of the function:$f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$ The domain of the function is $-1\leq x \leq 1$ $f(-1)=(\sin^{-1}(-1))^2-(\cot^{-1}(-1))^2=\frac{\pi^2}{4}-\frac{9\pi^2}{16}=-\frac{-5\...
Maverick's user avatar
  • 9,599
0 votes
0 answers
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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 ...
Syamaprasad Chakrabarti's user avatar