Inverse Trigonometric Functions

TrigonometryMATH 103

S. Rook

Overview

• Section 4.7 in the textbook:– Review of inverse functions– Inverse sine function– Inverse cosine function– Inverse tangent function– Inverse trigonometric functions and right triangles

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Review of Inverse Functions

Review of Inverse Functions

• Graphically, a function f has an inverse if it passes the horizontal line test

f is said to be one-to-one• Given a function f, let f-1 be the relation that results

when we swap the x and y coordinates for each point in f

• If f and f-1 are inverses, their domains and ranges are interchanged:– i.e. the domain of f becomes the range of f-1 & the range

of f becomes the domain of f-1 and vice versa

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Review of Inverse Functions (Continued)

• None of the six trigonometric functions have inverses as they are currently defined– All fail the horizontal line test– We will examine how to solve this problem soon

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Inverse Sine Function

Inverse Sine Function

• As mentioned earlier, y = sin x has no inverse because it fails the horizontal line test

• However, if we RESTRICT the domain of y = sin x, we can force it to be one-to-one– A common domain

restriction is– The restricted domain now passes the horizontal line test

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22

x

Inverse Sine Function (Continued)

• The inverse function of y = sin x is y = sin-1 x– Switch all (x, y) pairs in

the restricted domain of y = sin x

– A COMMON MISTAKE is to confuse the inverse notation with the reciprocal• To avoid confusion, y = sin-1 x is often written as y =

arcsin x– Pronounced “arc sine”– Be familiar with BOTH notations

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xxsin

1sin 1

Inverse Sine Function (Continued)

• For the restricted domain of y = sin x: D: [-π⁄2, π⁄2]; R: [-1, 1]

• Then for y = arcsin x: D: [-1, 1]; R: [- π⁄2, π⁄2]

– Recall that functions and their inverses swap domain and range

– This corresponds to angle in either QI or QIV

y = sin-1 x and y = arcsin x both mean x = sin y– i.e. y is the angle in the interval [- π⁄2, π⁄2]

whose sine is x

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Inverse Sine Function (Example)

Ex 1: Evaluate if possible without using a calculator – leave the answer in radians:

a)

b) arcsin(-2)

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2

1sin 1

Inverse Sine Function (Example)

Ex 2: Evaluate if possible using a calculator – leave the answer in degrees:

11

4664.0sin 1

Inverse Cosine Function

Inverse Cosine Function

• As mentioned earlier, y = cos x has no inverse because it fails the horizontal line test

• However, if we RESTRICT the domain of y = cos x, we can force it to be one-to-one– A common domain

restriction is– The restricted domain now passes the horizontal line test

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x0

Inverse Cosine Function (Continued)

• The inverse function of y = cos x is y = cos-1 x– Switch all (x, y) pairs in

the restricted domain of y = cos x

– To avoid confusion, y = cos-1 x is often written as y = arccos x• Pronounced “arc

cosine”• Be familiar with BOTH notations

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Inverse Cosine Function (Continued)

• For the restricted domain of y = cos x: D: [0, π]; R: [-1, 1]

• Then for y = arccos x: D: [-1, 1]; R: [0, π]

– This corresponds to an angle in either QI or QII

y = cos-1 x and y = arccos x both mean x = cos y– i.e. y is the angle in the interval [0, π]

whose cosine is x15

Inverse Cosine Function (Example)

Ex 3: Evaluate if possible without using a calculator – leave the answer in radians:

a) arccos(-3⁄2)

b) cos-1(1)

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Inverse Tangent Function

Inverse Tangent Function

• As mentioned earlier, y = tan x has no inverse because it fails the horizontal line test

• However, if we RESTRICT the domain of y = tan x, we can force it to be one-to-one– A common domain

restriction is– The restricted domain now passes the horizontal line test

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22

x

Inverse Tangent Function (Continued)

• The inverse function of y = tan x is y = tan-1 x– Switch all (x, y) pairs in

the restricted domain of y = tan x

– To avoid confusion, y = tan-1 x is often written as y = arctan x• Pronounced “arc

tangent”• Be familiar with BOTH notations

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Inverse Tangent Function (Continued)

• For the restricted domain of y = tan x: D: [-π⁄2, π⁄2]; R: (-oo, +oo)

• Then for y = arctan x: D: (-oo, +oo); R: [-π⁄2, π⁄2]

– This corresponds to an angle in either QI or QIV

y = tan-1 x and y = arctan x both mean x = tan y– i.e. y is the angle in the interval [-π⁄2, π⁄2]

whose tangent is x20

Inverse Tangent Function (Example)

Ex 4: Evaluate if possible without using a calculator – leave the answer in radians:

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1arctan

Inverse Trigonometric Functions and Right Triangles

Taking the Inverse of a Function

• Recall what happens when we take the inverse of a function:

• e.g. Given x = 3, because y = ln x and ex are inverses:

– In other words, we get the original argument PROVIDED that the argument lies in the domain of the function AND its inverse

– This also applies to the trigonometric functions and their inverse trigonometric functions

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3ln 3ln3 ee

xxffxff 11

Inverse Trigonometric Functions and Right Triangles

• The same technique does not work when the functions are NOT inverses– E.g. tan(sin-1 x)

• Recall the meaning of sin-1 x

• i.e. the sine of what angle results in x

• With this information, we can construct a right triangle using Definition II of the Trigonometric functions– We can use the right triangle to find

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xxx sinsinsin 1

tansintan 1 x

Inverse Trigonometric Functions and Right Triangles (Example)

Ex 5: Evaluate without using a calculator:

a) b)

c) d)

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2

1sinsin 1

4

3tancsc 1

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1cossin 1

6

7coscos 1

Inverse Trigonometric Functions and Right Triangles (Example)

Ex 6: Write an equivalent expression that involves x only – assume x is positive:

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x

1sinsec 1

Summary• After studying these slides, you should be able to:– State whether or not an argument falls in the domain

of the inverse sine, inverse cosine, or inverse tangent– Evaluate the inverse trigonometric functions both by

hand or by calculator– Evaluate expressions using inverse trigonometric

functions• Additional Practice– See the list of suggested problems for 4.7

• Next lesson– Proving Identities (Section 5.1)

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