4.1: Basic Inverse Trigonometric Functions
Introduction
Recall that an inverse function is a reflection of the function over the line
Learning Objectives
 Understand and evaluate inverse trigonometric functions.
 Extend the inverse trigonometric functions to include the
csc−1,sec−1 andcot−1 functions.  Apply inverse trigonometric functions to the critical values on the unit circle.
Defining the Inverse of the Trigonometric Ratios
Recall from Chapter 1, the ratios of the six trig functions and their inverses, with regard to the unit circle.
These ratios can be used to find any
Example 1: Find the measure of the angles below.
a.
b.
Solution: For part a, you need to use the sine function and part b utilizes the tangent function. Because both problems require you to solve for an angle, the inverse of each must be used.
a.
b.
The trigonometric value
Example 2: Find the angle,
Solution: The
Recall that inverse trigonometric functions are also used to find the angle of depression or elevation.
Example 3: A new outdoor skating rink has just been installed outside a local community center. A light is mounted on a pole 25 feet above the ground. The light must be placed at an angle so that it will illuminate the end of the skating rink. If the end of the rink is 60 feet from the pole, at what angle of depression should the light be installed?
Solution: In this diagram, the angle of depression, which is located outside of the triangle, is not known. Recall, the angle of depression equals the angle of elevation. For the angle of elevation, the pole where the light is located is the opposite and is 25 feet high. The length of the rink is the adjacent side and is 60 feet in length. To calculate the measure of the angle of elevation the trigonometric ratio for tangent can be applied.
The angle of depression at which the light must be placed to light the rink is
Exact Values for Inverse Sine, Cosine, and Tangent
Recall the unit circle and the critical values. With the inverse trigonometric functions, you can find the angle value (in either radians or degrees) when given the ratio and function. Make sure that you find all solutions within the given interval.
Example 4: Find the exact value of each expression without a calculator, in
a.
b.
c.
Solution: These are all values from the special right triangles and the unit circle.
a. Recall that
b.
c.
Notice how each one of these examples yield two answers. This poses a problem when finding a singular inverse for each of the trig functions. Therefore, we need to restrict the domain in which the inverses can be found, which will be addressed in the next section. Unless otherwise stated, all angles are in radians.
Finding Inverses Algebraically
In the Prerequisite Chapter, you learned that each function has an inverse relation and that this inverse relation is a function only if the original function is onetoone. A function is onetoone when its graph passes both the vertical and the horizontal line test. This means that every vertical and horizontal line will intersect the graph in exactly one place.
This is the graph of
First, switch
Next, multiply both sides by
Then, apply the distributive property and put all the
Divide by
Finally, multiply the right side by
Therefore the inverse of
The symbol
Example 5: Find the inverse of
Solution: To find the inverse algebraically, switch
Example 6: Find the inverse of
Solution:
a.
Example 7: Find the inverse of the trigonometric function
Solution:
Points to Consider
 What is the difference between an inverse and a reciprocal?
 Considering that most graphing calculators do not have
csc,sec orcot buttons, how would you find the inverse of each of these?  Besides algebraically, is there another way to find the inverse?
Review Questions
 Use the special triangles or the unit circle to evaluate each of the following:

cos120∘ 
csc3π4 
tan5π3

 Find the exact value of each inverse function, without a calculator in
[0,2π) :
cos−1(0) 
tan−1(−3√) 
sin−1(−12)

Find the value of the missing angle.
 What is the value of the angle with its terminal side passing through (14, 23)?
 A 9foot ladder is leaning against a wall. If the foot of the ladder is 4 feet from the base of the wall, what angle does the ladder make with the floor?
Find the inverse of the following functions.

f(x)=2x3−5 
y=13tan−1(34x−5) 
g(x)=2sin(x−1)+4 
h(x)=5−cos−1(2x+3)
Review Answers


−12 
2√ 
−3√

π2,3π2 
2π3,5π3 
11π6,7π6


cosθ=1217→cos−11217=45.1∘ 
sinθ=2536→sin−13136=59.44∘  This problem uses tangent inverse.
tanx=−23−14→x=tan−12314=58.67∘ (value graphing calculator will produce). However, this is the reference angle. Our angle is in the third quadrant because both thex andy values are negative. The angle is180∘+58.67∘=238.67∘ . 
cosAcos−149∠A=49=A=63.6∘ 
f(x)yxx+5x+52x+52−−−−−√3=2x3−5=2x3−5=2y3−5=2y3=y3=y 
yx3xtan(3x)tan(3x)+54(tan(3x)+5)3=13tan−1(34x−5)=13tan−1(34y−5)=tan−1(34y−5)=34y−5=34y=y 
g(x)yxx−4x−42sin−1(x−42)1+sin−1(x−42)=2sin(x−1)+4=2sin(x−1)+4=2sin(y−1)+4=2sin(y−1)=sin(y−1)=y−1=y 
h(x)yxx−55−xcos(5−x)cos(5−x)−3cos(5−x)−32=5−cos−1(2x+3)=5−cos−1(2x+3)=5−cos−1(2y+3)=−cos−1(2y+3)=cos−1(2y+3)=2y+3=2y=y
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