7.5

RIGHT TRIANGLES: INVERSE TRIGONOMETRIC

FUNCTIONS

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

The Inverse Sine Function

Example 2Use the inverse sine function to find the angles in the figure.

SolutionUsing our calculator’s inverse sine function:

sin θ = 3/5 = 0.6 so θ = sin−1(0.6) = 36.87◦ sin φ = 4/5 = 0.8 so φ = sin−1(0.8) = 53.13◦

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

For 0 ≤ x ≤ 1: arcsin x = sin−1 x = The angle in a right triangle whose sine is x.

θ

3

4

5 φ

The Inverse Tangent Function

Example 3The grade of a road is 5.8%. What angle does the road make with the horizontal?SolutionSince the grade is 5.8%, the road climbs 5.8 feet for 100 feet; see the figure. We see that tan θ = 5.8/100 = 0.058.So θ = tan−1(0.058) = 3.319◦

using a calculator.

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

arctan x = tan−1 x = The angle in a right triangle whose tan is x.

θ

5.8 ft

100 ft

A road rising at a grade of 5.8% (not to scale)

Summary of Inverse Trigonometric Functions

We define:• the arc sine or inverse sine function as arcsin x = sin−1 x = The angle in a right triangle whose sine is x• the arc cosine or inverse cosine function as arccos x = cos−1 x = The angle in a right triangle whose cosine is x• the arc tangent or inverse tangent function as arctan x = tan−1 x = The angle in a right triangle whose tangent is x.This means that for an angle θ in a right triangle (other than the right angle),

sin θ = x means θ = sin−1 xcos θ = x means θ = cos−1 xtan θ = x means θ = tan−1 x.

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally