## 17Calculus Precalculus - Inverse Trig Functions

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This page covers inverse trigonometric functions specifically. Inverses of functions in general are discussed on the inverse functions page.

Plot 1 - $$\sin(x)$$

As mentioned on the inverse functions page, the trig functions do not pass the horizontal line test and, therefore, do not, in general, have inverses. However, in order to get inverses, we restrict the domain. This allows us to talk about inverses when we have trig functions.

For example, if we have the trig function $$f(x)=\sin x$$, we can restrict the domain and end up with a one-to-one function. This is shown in plot 1. The blue line is $$y=\sin x$$. If we restrict the domain to $$-\pi/2 \leq x \leq \pi/2$$, we end up with the red part of $$y=\sin x$$. This part passes the horizontal line test and, therefore, has an inverse.

Here is a great video that goes into detail for several trig functions like we did above with $$y=\sin x$$.

### PatrickJMT - Inverse Trigonometric Functions, Part 1 ( Basic Introduction ) [10min-46secs]

video by PatrickJMT

Working with Inverse Trig Functions

Inverse trig functions are used so much in calculus that we have special notation to write them by putting arc in front.

function inverse $$\sin(x)$$ $$\sin^{-1}(x) =$$ $$\arcsin(x)$$ $$\cos(x)$$ $$\cos^{-1}(x) =$$ $$\arccos(x)$$

The other trig functions have similar notation. Of course, we assume that the domains of these functions are restricted so that they pass the horizontal line test, which allows the inverse trig function to make sense. We do not state it and most instructors and textbooks do not either. So you just need to remember it.

Since $$\arcsin(x)$$ is the inverse, we know that $$\sin(\arcsin(x)) = \arcsin(\sin(x)) = x$$. This works not just with x but with any expression. For example, $$\sin(\arcsin(72k)) = \arcsin(\sin(72k)) = 72k$$ and $$\sin(\arcsin(17)) = \arcsin(\sin(17)) = 17$$.

Here is a good video showing some examples working with inverse trig functions.

### PatrickJMT - Inverse Trigonometric Functions , Part 2 ( Evaluating Inverse Trig Functions ) [10min-59secs]

video by PatrickJMT

One type of problem that you will probably be asked to solve is to evaluate an expression like $$\sin(\arccos(x))$$. Notice that we have an inverse cosine inside of a sine, so you can't directly write x. To solve an expression like this, we use the technique of substitution. See the practice problems for explanation on how to do this.

Practice

Instructions - - Unless otherwise instructed, evaluate (simplify) the given function without using a calculator and give your answers in exact, simplified form.

Unless otherwise instructed, evaluate (simplify) $$\sin( \arccos( 5 / 13 ) )$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\sin( \arccos( 5 / 13 ) )$$ without using a calculator and give your answer in exact, simplified form.

$$\sin(\arccos(5 / 13)) = 12 / 13$$

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\sin( \arccos( 5 / 13 ) )$$ without using a calculator and give your answer in exact, simplified form.

Solution

We start on the inside by letting $$\theta = \arccos(5 / 13)$$. Taking the cosine of each side, we get $$\cos(\theta) = 5 / 13$$.
From this equation, we set up the triangle on the right, which means $$b = 5$$ and $$c = 13$$.

Using the Pythagorean Theorem, we know that $$c^2 = a^2 + b^2$$ or $$13^2 = a^2 + 5^2$$. Solving for $$a$$ we get $$a^2 = 13^2 - 5^2 = 169 - 25 = 144$$ or $$a = \pm 12$$. We choose $$a = +12$$.

Now that we have a triangle with the length of all sides known, we can write $$\sin( \arccos(5 / 13)) = \sin(\theta)$$ and use the triangle to get $$\sin(\theta) = 12 / 13$$.

$$\sin(\arccos(5 / 13)) = 12 / 13$$

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Evaluate $$\sin^{-1}(1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin^{-1}(1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3002 video

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Evaluate $$\sin^{-1}(\sqrt{3}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin^{-1}(\sqrt{3}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3003 video

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Evaluate $$\sin^{-1}(-1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin^{-1}(-1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3004 video

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Evaluate $$\sin^{-1}(-\sqrt{2}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin^{-1}(-\sqrt{2}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3005 video

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Evaluate $$\cos^{-1}(1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\cos^{-1}(1/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3006 video

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Evaluate $$\cos^{-1}(-\sqrt{3}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\cos^{-1}(-\sqrt{3}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3007 video

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Evaluate $$\cos^{-1}(-\sqrt{2}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\cos^{-1}(-\sqrt{2}/2)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3008 video

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Evaluate $$\tan^{-1}(0)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}(0)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3009 video

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Evaluate $$\tan^{-1}(1)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}(1)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3010 video

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Evaluate $$\tan^{-1}(-1)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}(-1)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3011 video

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Evaluate $$\tan^{-1}(\sqrt{3})$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}(\sqrt{3})$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3012 video

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Evaluate $$\tan^{-1}(-\sqrt{3}/3)$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}(-\sqrt{3}/3)$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3013 video

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Evaluate $$\sin^{-1}[\cos(\pi/3)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin^{-1}[\cos(\pi/3)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3014 video

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Evaluate $$\tan^{-1}[\sin(\pi/2)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan^{-1}[\sin(\pi/2)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3015 video

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Evaluate $$\cos^{-1}[\tan \pi]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\cos^{-1}[\tan \pi]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3016 video

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Evaluate $$\sin[\cos^{-1}(3/5)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sin[\cos^{-1}(3/5)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3017 video

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Evaluate $$\cos[\tan^{-1}(8/15)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\cos[\tan^{-1}(8/15)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3018 video

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Evaluate $$\tan[\sin^{-1}(-5/13)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\tan[\sin^{-1}(-5/13)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3019 video

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Evaluate $$\sec[\sin^{-1}(1/4)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\sec[\sin^{-1}(1/4)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3020 video

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Evaluate $$\csc[\cos^{-1}(-7/9)]$$ without using a calculator, giving your answer in exact terms in radians.

Problem Statement

Evaluate $$\csc[\cos^{-1}(-7/9)]$$ without using a calculator, giving your answer in exact terms in radians.

Solution

### 3021 video

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Unless otherwise instructed, evaluate (simplify) $$\cos(\arcsin(1/2))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\cos(\arcsin(1/2))$$ without using a calculator and give your answer in exact, simplified form.

Solution

### 1561 video

video by PatrickJMT

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Unless otherwise instructed, evaluate (simplify) $$\tan(\arcsin(2/3))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\tan(\arcsin(2/3))$$ without using a calculator and give your answer in exact, simplified form.

Solution

### 1562 video

video by PatrickJMT

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Unless otherwise instructed, evaluate (simplify) $$\sin(2\arctan(\sqrt{2}))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\sin(2\arctan(\sqrt{2}))$$ without using a calculator and give your answer in exact, simplified form.

Solution

### 1563 video

video by PatrickJMT

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Unless otherwise instructed, evaluate (simplify) $$\cos( \arcsin(x))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\cos( \arcsin(x))$$ without using a calculator and give your answer in exact, simplified form.

$$\cos( \arcsin(x)) = \sqrt{1-x^2}$$

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\cos( \arcsin(x))$$ without using a calculator and give your answer in exact, simplified form.

Solution

Starting on the inside, we have $$\theta = \arcsin(x)$$. Taking the sine of both sides yields $$\sin(\theta) = x$$. We use this equation to set up the triangle to the right, $$a = x$$ and $$c = 1$$.

Now we use the Pythagorean Theorem to solve for b.
$$\begin{array}{rcl} x^2 + b^2 & = & 1 \\ b^2 & = & 1 - x^2 \\ b & = & \pm \sqrt{1-x^2} \end{array}$$

We choose the positive square root.

Now we can write $$\cos(\theta) = \sqrt{1-x^2}$$.

$$\cos( \arcsin(x)) = \sqrt{1-x^2}$$

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Unless otherwise instructed, evaluate (simplify) $$\sec( \arctan(x/3))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\sec( \arctan(x/3))$$ without using a calculator and give your answer in exact, simplified form.

$$\displaystyle{\sec(\arctan(x/3))=\frac{\sqrt{x^2+9}}{3}}$$

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\sec( \arctan(x/3))$$ without using a calculator and give your answer in exact, simplified form.

Solution

We start on the inside by letting $$\theta = \arctan(x/3)$$. Take the tangent of both sides to get $$\tan(\theta) = x/3$$. We use this equation to set up the triangle to the right, $$a = x, ~~ b = 3$$. Using the Pythagorean Theorem, we can find $$c$$, i.e. $$c^2 = x^2 + 3^2 ~~ \to ~~ c = \pm \sqrt{x^2 + 9}$$.

We choose the positive square root.
Now we have $$\displaystyle{ \sec(\theta) = \frac{\sqrt{x^2+9}}{3} }$$

$$\displaystyle{\sec(\arctan(x/3))=\frac{\sqrt{x^2+9}}{3}}$$

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Evaluate $$\sin[\cos^{-1}(x)]$$ without using a calculator, giving your answer in terms of x.

Problem Statement

Evaluate $$\sin[\cos^{-1}(x)]$$ without using a calculator, giving your answer in terms of x.

Solution

### 3022 video

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Evaluate $$\tan[\sin^{-1}(2x)]$$ without using a calculator, giving your answer in terms of x.

Problem Statement

Evaluate $$\tan[\sin^{-1}(2x)]$$ without using a calculator, giving your answer in terms of x.

Solution

### 3023 video

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Evaluate $$\cos[\tan^{-1}(1/x)]$$ without using a calculator, giving your answer in terms of x.

Problem Statement

Evaluate $$\cos[\tan^{-1}(1/x)]$$ without using a calculator, giving your answer in terms of x.

Solution

### 3024 video

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Evaluate $$\sin[\arctan(u/3)]$$ expressing your answer in terms of $$u$$.

Problem Statement

Evaluate $$\sin[\arctan(u/3)]$$ expressing your answer in terms of $$u$$.

Solution

### 3026 video

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Evaluate $$\displaystyle{ \csc \left[ \cos^{-1} \left( \frac{x}{\sqrt{x^2+9}} \right) \right] }$$ without using a calculator, giving your answer in terms of x.

Problem Statement

Evaluate $$\displaystyle{ \csc \left[ \cos^{-1} \left( \frac{x}{\sqrt{x^2+9}} \right) \right] }$$ without using a calculator, giving your answer in terms of x.

Solution

### 3025 video

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Unless otherwise instructed, evaluate (simplify) $$\tan(\arcsin(x))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\tan(\arcsin(x))$$ without using a calculator and give your answer in exact, simplified form.

Solution

### 1564 video

video by PatrickJMT

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Unless otherwise instructed, evaluate (simplify) $$\csc(\arctan(2x))$$ without using a calculator and give your answer in exact, simplified form.

Problem Statement

Unless otherwise instructed, evaluate (simplify) $$\csc(\arctan(2x))$$ without using a calculator and give your answer in exact, simplified form.

Solution

### 1565 video

video by PatrickJMT

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### inverse functions 17calculus youtube playlist

Really UNDERSTAND Precalculus

 basics of trig inverse functions

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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