Questions tagged [inverse-trigonometric-functions]

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

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
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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
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2 answers
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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
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3 answers
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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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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
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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
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1 answer
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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
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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
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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
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1 answer
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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 ...
J. Tanin's user avatar
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2 answers
67 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
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1 vote
3 answers
196 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
74 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
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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