1
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2answers
20 views

Clarification on the domain of $\arcsin(\sqrt{1-x^2})$

As the title says, I don't understand how to find the domain of $\arcsin(\sqrt{1-x^2})$. I kinda understand how it would equate to it would be -1 < x < 1 (inclusive of 1 and -1) by definition of ...
1
vote
2answers
43 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
1
vote
1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
1
vote
1answer
57 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
0
votes
4answers
95 views

Finding $\sin^{-1}(x)$ without using a calculator

I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the ...
0
votes
1answer
20 views

Inverting complicated function (possibly using secant root finder)

So I have the following equation from the 2002 paper "A Rapid Hierarchical Rendering Technique for Translucent Materials" http://graphics.ucsd.edu/~henrik/papers/fast_bssrdf/fast_bssrdf.pdf Here is ...
0
votes
2answers
43 views

Domain of arctan(1/x)

I had this as part of a question in an exam. And, I reasoned, even when it's arctan(1/0) (undefined), it is pi/2. And, so I said, domain belongs to all Real Numbers. Why isn't it this
1
vote
1answer
41 views

solve $x+\sin(x)=k$ for $x$ [duplicate]

This question has been proposed to me and thus far it has baffled me: $$ x + \sin(x) = k$$ solve for x. Another way of looking at it is find $f^{-1}(x)$ given that $f(x)=x + \sin(x)$. Wolfram alpha ...
2
votes
2answers
45 views

Is $\sec^{-1}(\sec(\pi/2)) = \pi/2$?

I think it shouldn't be defined as $\pi/2$ is not in the range of the function $\sec^{-1}(x)$ Wolfram confused me by giving the answer as $\pi/2$ : Link But it mentions on another page that $\pi/2$ ...
1
vote
1answer
22 views

Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$ So what is the logic behind simplifying expressions like this, ...
5
votes
1answer
77 views

How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to ...
1
vote
0answers
34 views

Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
0
votes
2answers
38 views

How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
0
votes
1answer
44 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
0
votes
1answer
49 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
0
votes
2answers
53 views

Find the range of arcsin$((1-x^2)^{0.5})$

Title says it all, how do you get the answer to this? So far I only reach $0<1-x^2<pi/2$ but I get an invalid answer from here. the correct answer is $0<x<pi/2$. Any help is appreciated, ...
1
vote
1answer
15 views

Arc Tangents and Equation

For one of the problems in my book, it requires you to put the arc tangent into the 2piK equation and solve for the arc tangents and lie in [0,2pi]. For: arctan(117)+piK the answers are 1.5622 and ...
1
vote
1answer
79 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
0
votes
3answers
106 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
2
votes
3answers
39 views

Please solve this in details inverse problem i am using complement angle formula

For any $x \in [-1,0) \cup (0,1]$, how can I prove that: $$\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x$$ Also, can someone explain to me how to understand the graphs of $sin$ and $cos$ functions?
0
votes
0answers
30 views

Inverse trig functions [duplicate]

What would be the inverse function of $f(x) =x \cot \frac{\pi }{x}$ on the interval $ x\geq 3$? I don't know what to do about that pesky x in the front of the problem. Otherwise, the problem would be ...
0
votes
2answers
57 views

Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
1
vote
3answers
34 views

Proves of identities in inverse trigonometry

Can someone please help me prove the following results from inverse trigonometry? $$\tan^{-1}x + \tan^{-1}y = \pi + \tan^{-1}\frac{x+y}{1-xy}( x>0, y>0, xy>1)$$ and $$\tan^{-1}x + ...
1
vote
1answer
18 views

Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
0
votes
1answer
29 views

Finding the integral of an inverse cosine function?

I've just been having trouble with this question: "Differentiate $xcos^{-1}x$ and hence find the integral of $cos^{-1}x$. Hint: Try using the substitution $u=1-x^2$." Finding the derivative wasn't ...
1
vote
1answer
44 views

Expressing an inverse trig function?

I just need a little help with this question: "Express cos$y$ in terms of cos $y/2$ and hence show that tan$^{-1} sqrt[(1-x)/(1+x)] = 1/2$ cos$^{-1}x$, for $0<x<1$." I can do the first part, ...
2
votes
4answers
335 views

calculator issue: radians or degrees for inverse trig

It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
1
vote
3answers
55 views

Solve inverse tangents

How do I solve the following equation: $$ \tan^{-1}\frac{x}{10^6}+\tan^{-1}\frac{x}{10^7}=90^{\circ}$$ WA Step by step solution from wolframalpha is unavailable.
2
votes
1answer
293 views

Trick: Substitution in inverse trigonometry.

My friends say, it is some what difficult to know, which trigonometric function has to be substituted in the inverse trigonometric equations, to get the correct solution. So, I thought to take up this ...
1
vote
2answers
273 views

Simplifying an inverse trig function?

I am trying to figure out how to simplify this expression but I am not quite sure how these inverses work. What sort of approach should I take for this equation? ...
1
vote
1answer
69 views

Calculate the inverse for $\arctan(x^2+1), x≥0$

I have no idea how to solve this problem. Calculate the inverse for the function: $$f(x) = \arctan(x^2+1),\quad x≥0 .$$ Also specify $D_{f^{-1}}$ and $V_{f^{-1}}$. I would really ...
0
votes
3answers
212 views

Find the domain and range of $y=\cos^{-1} \sqrt{1-x}$

Find the domain and range of $y=\cos^{-1}\sqrt{1-x}$. Can someone please help me with question above, as to how it's done? Thanks. I am unfamiliar with what you do when there is a square root.
4
votes
3answers
398 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\sin({\arccos{x}})=\sqrt{1-x^2}$$ $$\cos{\arcsin{x}}=\sqrt{1-x^2}$$ $$\sin{\arctan{x}}=\frac{x}{\sqrt{1+x^2}}$$ $$\cos{\arctan{x}}=\frac{1}{\sqrt{1+x^2}}$$ ...
2
votes
1answer
78 views

Simplify difference of two arc tangents?

I have a problem, that I am trying to simplify, but there does not seem to be something obvious regarding it. Very simply, I am trying to figure out if there is a way to 'open' the following: $$ ...
11
votes
5answers
2k views

What's the difference between arccos(x) and sec(x)

My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
0
votes
1answer
30 views

How can I solve an equation based off of a quadrant and equation form given an angle?

Given that 3pi/2 < z < 2pi x = arccos(sin(z)) Given different values for z (which are angles on the unit circle) how would I write the results in these two forms, where C is a constant?: a.) ...
2
votes
2answers
380 views

Finding ALL solutions to $2(\sin^2(x)) - 5\sin(x)-3 = 0$?

What does it mean to find "ALL possible solutions?" I know it has something to do with simplifying the equation, getting the angle (in radians) by doing the inverse.. and adding $2\pi n$? So given ...
0
votes
1answer
64 views

Did I solve all of the steps of this Trig question properly?

Thanks to some help from the community, I think I did this problem correctly, but I would like someone to confirm that I indeed do it right. Thanks. Question: Let $0 \le x \le 1$. (i.) Find the ...
1
vote
2answers
73 views

How do I write a trig function that includes inverses in terms of another variable?

It's been awhile since I've used trig and I feel stupid asking this question lol but here goes: Given: $z = \tan(\arcsin(x))$ Question: How do I write something like that in terms of $x$? Thanks! ...
0
votes
2answers
121 views

Graph of an inverse trig function.

Which of the following is equivalent to the graph of $arcsin(x)$ ? (a) Reflecting $arccos(x)$ about the y-axis, then shift down by $\pi /2$ units. (b) Reflecting $arccos(x)$ about the x-axis, then ...
0
votes
1answer
44 views

Proving the area function has an inverse

I am able to differentiate A at x using the FTC, but then I was wondering how one could show that A was one to one and prove that it has an inverse. If anybody could please help.
2
votes
2answers
348 views

Trigonometric general solution to ordinary differential equation

Solve: $$\frac{dx}{dy}=(x^{2}-x-12)(1+\tan^{2}{y})$$ This is a first order, linear, separable ODE, so it can be solved by rearranging to: $$\frac{dx}{x^{2}-x-12}=(1+\tan^{2}{y})\:dy$$ And then ...
1
vote
3answers
875 views

Prove $ \sqrt{\arctan(x)} = (1/2) \arccos((1-x)/(1+x))$

$$ \sqrt{\arctan(x)} = \dfrac{1}{2} \arccos\left(\dfrac{1-x}{1+x}\right)$$ I have been trying to solve this problem for the past hour, but I'm not able to solve it as I have just started solving ...
2
votes
2answers
335 views

Write the above function in simplest form.

$$\tan^{-1}\left(\dfrac{3a^2x-x^3}{a^3-3ax^2}\right),\;a>0;\; -\frac{a}{\sqrt{3}}\leqslant x \leqslant \frac{a}{\sqrt{3}}$$ Hi, Please help me to solve this problem. As I can solve simple ...
1
vote
1answer
433 views

Write the following functions in simplest form.

$$ \arctan\left(\frac{x}{\sqrt{a^2-x^2}}\right)$$ Hi, I am not able to solve this problem from last 1 hour. Please help me to solve this question. As I can solve simple inverse ...
2
votes
3answers
766 views

Simplify $\tan^{-1}[(\cos x - \sin x)/(\cos x + \sin x)]$

Write the following functions in simplest form: $$\tan^{-1}\left(\frac{\cos(x)-\sin(x)}{\cos(x)+\sin(x)}\right), \quad 0<x<\pi$$ Please help me to solve this problem. I have been trying ...
8
votes
1answer
261 views

$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$

I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for ...
1
vote
2answers
492 views

Simplify $\arcsin(\sin(x))$ when $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$

Simplify $\arcsin(\sin(x))$ when $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$ I realize that $\arcsin(\theta)$ is restriced to $\frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2}$ in order to be one to ...
1
vote
2answers
166 views

Simple Trig inverse problem

Suppose we have $x=\sin a \cos b, y=\sin a\sin b, z=\cos a$. I want to invert them to get $a,b$ in terms of $x,y,z$. At first sight it appears very simple, for example my first reaction would be to ...
2
votes
1answer
762 views

Differentiate an Inverse Function, Two Methods?

I would like to take the derivative of this inverse function at $\pi$: $f(x) = 2x + \cos{x}$, given that ${f}^{-1}(\pi) = \frac{\pi}{2}$. I know that there are two methods of doing it. Let me ...