# 8.3 Inverse trigonometric functions  (Page 3/15)

 Page 3 / 15

Evaluate each of the following.

1. ${\text{sin}}^{-1}\left(-1\right)$
2. ${\mathrm{tan}}^{-1}\left(-1\right)$
3. ${\mathrm{cos}}^{-1}\left(-1\right)$
4. ${\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)$

a. $\text{\hspace{0.17em}}-\frac{\pi }{2};\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}-\frac{\pi }{4};\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}\pi ;\text{\hspace{0.17em}}$ d. $\text{\hspace{0.17em}}\frac{\pi }{3}\text{\hspace{0.17em}}$

## Using a calculator to evaluate inverse trigonometric functions

To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN-1, ARCSIN, or ASIN.

In the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.

In these examples and exercises, the answers will be interpreted as angles and we will use $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.

## Evaluating the inverse sine on a calculator

Evaluate $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(0.97\right)\text{\hspace{0.17em}}$ using a calculator.

Because the output of the inverse function is an angle, the calculator will give us a degree value if in degree mode and a radian value if in radian mode. Calculators also use the same domain restrictions on the angles as we are using.

In radian mode, $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(0.97\right)\approx 1.3252.\text{\hspace{0.17em}}$ In degree mode, $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(0.97\right)\approx 75.93°.\text{\hspace{0.17em}}$ Note that in calculus and beyond we will use radians in almost all cases.

Evaluate $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(-0.4\right)\text{\hspace{0.17em}}$ using a calculator.

1.9823 or 113.578°

Given two sides of a right triangle like the one shown in [link] , find an angle.

1. If one given side is the hypotenuse of length $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ and the side of length $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ adjacent to the desired angle is given, use the equation $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta ={\mathrm{cos}}^{-1}\left(\frac{a}{h}\right).$
2. If one given side is the hypotenuse of length $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ and the side of length $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ opposite to the desired angle is given, use the equation $\text{\hspace{0.17em}}\theta ={\mathrm{sin}}^{-1}\left(\frac{p}{h}\right).$
3. If the two legs (the sides adjacent to the right angle) are given, then use the equation $\text{\hspace{0.17em}}\theta ={\mathrm{tan}}^{-1}\left(\frac{p}{a}\right).$

## Applying the inverse cosine to a right triangle

Solve the triangle in [link] for the angle $\text{\hspace{0.17em}}\theta .$

Because we know the hypotenuse and the side adjacent to the angle, it makes sense for us to use the cosine function.

Solve the triangle in [link] for the angle $\text{\hspace{0.17em}}\theta .$

${\mathrm{sin}}^{-1}\left(0.6\right)=36.87°=0.6435\text{\hspace{0.17em}}$ radians

## Finding exact values of composite functions with inverse trigonometric functions

There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ be two different trigonometric functions belonging to the set $\text{\hspace{0.17em}}\left\{\mathrm{sin}\left(x\right),\mathrm{cos}\left(x\right),\mathrm{tan}\left(x\right)\right\}\text{\hspace{0.17em}}$ and let $\text{\hspace{0.17em}}{f}^{-1}\left(y\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{g}^{-1}\left(y\right)$ be their inverses.

#### Questions & Answers

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6x
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