Inverse Trigonometric Functions Trigonometry MATH 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

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

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

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

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

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

Inverse Sine Function (Example) Ex 1: Evaluate if possible without using a calculator – leave the answer in radians: a) b) arcsin(-2)

Inverse Sine Function (Example) Ex 2: Evaluate if possible using a calculator – leave the answer in degrees:

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

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

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 x

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)

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

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

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 x

Inverse Tangent Function (Example) Ex 4: Evaluate if possible without using a calculator – leave the answer in radians:

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

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

Inverse Trigonometric Functions and Right Triangles (Example) Ex 5: Evaluate without using a calculator: a) b) c) d)

Inverse Trigonometric Functions and Right Triangles (Example) Ex 6: Write an equivalent expression that involves x only – assume x is positive:

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)