# What is Inverse trigonometry: Definition and 14 Discussions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

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1. ### Cannot understand what ##(mod ~ \pi)## means in the given formula

My understanding is that it probably means either ##+ \pi## or ## - \pi##, so that ##\pi## is either added to or subtracted from the first term, just like ##|x| = \pm x##.
2. ### Find the domain of arccsc(e^2x)

Homework Statement Hello! Seems I have forgotten how to work with such values. Please, help me to find the correct approach to this type of exercises. Homework Equations The task is to find the domain of the function: f(x) = arccsc(e2x) The Attempt at a Solution The answer in the book is [0...
3. ### I Inverse hyperbolic function expressed as inverse trigonometry function

Consider ##y=\cos{-x}=\cos x=\cosh ix##. Thus, ##\pm x=\cos^{-1}y## and ##ix=\cosh^{-1}y##. So ##\cosh^{-1}y=\pm i\cos^{-1}y##. Renaming the variable ##y##, we have ##\cosh^{-1}x=\pm i\cos^{-1}x##. Next, we evaluate the derivative of ##\cosh^{-1}x## by converting it to ##\cos^{-1}x## using...
4. ### Proving a Trigonometric Identity Involving Inverse Functions

Homework Statement After having struggled yesterday with this as much as I could, I am posting this problem here- if ##ax+bsec(tan^{-1}x)=c## and ##ay+bsec(tan^{-1}y)=c##, then prove that ##\frac{x+y}{1-xy}=\frac{2ac}{a^2-c^2}##.Homework EquationsThe Attempt at a Solution My attempt-...
5. ### Inverse tangents in cyclic order

Homework Statement The problem- if $$\theta= tan^{-1}(\frac{a(a+b+c)}{bc})+tan^{-1}(\frac{b(a+b+c)}{ac})+tan^{-1}(\frac{c(a+b+c)}{ab})$$ , then find $$tan\theta$$ Homework EquationsThe Attempt at a Solution I tried to use these as sides of a triangle and use their properties, but other than...
6. ### Converting one inverse trig function to another

Homework Statement Express $$2 arctan (\sqrt\frac{a-b}{a+b} tan (\theta/2))$$ in terms of inverse cosine Homework Equations I realize it amounts to find a smart substitution, but I can't find one. The Attempt at a Solution I tried ##b/a=tan \theta## , but I can't find any way to get rid of...
7. ### What is the solution to the inverse trigonometry problem with a given range?

Homework Statement If 1/2<=x<=1 , then cos-1[ x/2 + (√(3-3x2))/2] + cos-1 x is equal to? Homework EquationsThe Attempt at a Solution Substituting x=cosθ = cos-1(cosθ) + cos-1 [ cos(θ)/2 + (√(3(1-cos2(θ)))/2] = θ + cos-1 [cos (π/3) cos θ + sin (π/3) sinθ] =...
8. ### MHB Derivatives and Inverse Trigonometry

Hey guys, I have a couple of questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, using inverse trigonometric derivative identities should work, right? I got y' = 1/sinØ + 1/cosØ and multiplied by the common...
9. ### Inverse trigonometry - Finding solutions

Homework Statement Which of the following is the solution set of the equation $$2\arccos(x)=\text{arccot}\left(\frac{2x^2-1}{2x\sqrt{1-x^2}}\right)$$ A)(0,1) B)(-1,1)-{0} C)(-1,0) D)[-1,1] Ans: A Homework Equations The Attempt at a Solution I start by rewriting LHS in terms of ##\arctan##...
10. ### Infinite series - Inverse trigonometry

Homework Statement The sum of the infinite terms of the series \text{arccot}\left(1^2+\frac{3}{4}\right)+\text{arccot}\left(2^2+\frac{3}{4}\right)+\text{arccot}\left(3^2+\frac{3}{4}\right)+... is equal to A)arctan(1) B)arctan(2) C)arctan(3) D)arctan(4) Ans: B Homework Equations The Attempt at...
11. ### Inverse trigonometry prove this

Homework Statement If cos^{-1}\frac{x}{a}+cos^{-1}\frac{y}{b} = \alpha then show that \frac{x^2}{a^2}-\frac{2xy}{ab}cos \alpha + \frac{y^2}{b^2} = sin^2 \alpha Homework Equations The Attempt at a Solution I assume inverse functions to be θ and β respectively. So in the second equation...
12. ### Inverse Trigonometry question

Homework Statement cos(2tan-1(1/7)) equals- a)sin(4cot-13) b)sin(3cot-14) c)cos(3cot-14) d)sin(4cot-14) Homework Equations The Attempt at a Solution I am completely stuck in this one. I tried using the formula 2tan^{-1}x=tan^{-1}(\frac{2x}{1-x^2}). Using this formula i got...
13. ### How to solve this inverse trigonometry question

Homework Statement tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3 2. The attempt at a solution tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3 tan-1[2x/(1-x2)]=2π/6 take tan on both sides 2x/(1-x2) =sqrt(3) quadratic equation so it should have 2 solutions(sqrt(3) and sqrt(1/3)).But this question...
14. ### Simple Inverse trigonometry question

This is a simple question but I really need to know why: why is the alternate form of arccosh(x) = log(x + sqrt(x^2 - 1)) and not log(x - sqrt(x^2 - 1)) How do I justify the plus/minus sign? I need to know this ASAP, thanks.