Lesson 4.7 Inverse Trigonometric Functions Essential Question: How do you evaluate and graph the inverses of trigonometric functions?

Before we start… Find 𝑓 𝜋 for 𝑓 𝑥 =3 sin 𝑥

What do you remember about inverse functions? We have know that for a function to have an inverse function, it must be one-to-one—that is, it must pass the Horizontal Line Test

If you notice, these trigonometric functions will fail the Horizontal Line test. In order to create inverse functions, you have to restrict the domain so that you only look at a small piece of the function.

Definition of Inverse Trigonometric Functions

How do you evaluate inverse trigonometric functions? You are looking for the angle that gives the ratio of sides. Use reference triangles and function graphs to help you.

If possible, find the exact value. arcsin (– 1)

If possible, find the exact value. sin −1 1 2

If possible, find the exact value. sin −1 3

Find the exact value. arcsin 3 2

Find the exact value. cos −1 (−0.5)

Find the exact value. arctan 1

Find the exact value. tan −1 3 3

Use a calculator to approximate the value, if possible. arctan 4.84

Use a calculator to approximate the value, if possible. arccos (−0.349)

Use a calculator to approximate the value, if possible. sin −1 (−1.1)

How do you graph inverse trigonometric functions? Recognize the characteristics of these functions including domain and range to graph. Intercepts Asymptotes

Library of Parent Functions: Inverse Trigonometric Functions Graph of 𝑓 𝑥 = arcsin 𝑥 Domain: −1,1 Range: − 𝜋 2 , 𝜋 2 Intercept: 0,0 Odd function Origin symmetry

Library of Parent Functions: Inverse Trigonometric Functions Graph of 𝑓 𝑥 = arccos 𝑥 Domain: −1,1 Range: 0,𝜋 y-intercept: 0, 𝜋 2

Library of Parent Functions: Inverse Trigonometric Functions Graph of 𝑓 𝑥 = arctan 𝑥 Range: − 𝜋 2 , 𝜋 2 Intercept: 0,0 Horizontal asymptotes: 𝑦=± 𝜋 2 Odd function Origin symmetry

Sketch a graph of y = arcsin x.

Sketch a graph of y = arcsin 2x.

Sketch a graph of 𝑦= cos −1 𝑥 .

Compare the graph of each function with the graph of 𝑓 𝑥 = arcsin 𝑥 .

Compare the graph of each function with the graph of 𝑓 𝑥 = arcsin 𝑥 .

Composition of Functions For all x in the domains of f and f – 1, inverse functions have the properties 𝑓 𝑓 −1 𝑥 =𝑥 and 𝑓 −1 𝑓 𝑥 =𝑥.

Inverse Properties If −1≤𝑥≤1 and − 𝜋 2 ≤𝑦≤ 𝜋 2 , then sin arcsin 𝑥 =𝑥 and arcsin sin 𝑦 =𝑦 . If −1≤𝑥≤1 and 0≤𝑦≤𝜋, then cos arccos 𝑥 =𝑥 and arccos cos 𝑦 =𝑦 . If x is a real number and − 𝜋 2 ≤𝑦≤ 𝜋 2 , then tan arctan 𝑥 =𝑥 and arctan tan 𝑦 =𝑦 .

If possible, find the exact value. tan arctan −14

If possible, find the exact value. cos arccos 0.54

If possible, find the exact value. arcsin sin 5𝜋 3

Find the exact value. cos arcsin − 3 5

Find the exact value. cos arctan − 3 4

Find the exact value. sin arccos 2 3

Find the exact value. tan arccos 2 3

Write each of the following as an algebraic expression in x. sec arctan 𝑥

Write each of the following as an algebraic expression in x. tan arccos 2𝑥

How do you evaluate and graph the inverses of trigonometric functions?

Ticket Out the Door Evaluate cot arcsin 5 6