1. Inverse Trig Functions
Lesson 3.5

2. Start with Sine Functionx
y = sin(x)
0.0000
0.5236
0.5000
1.0472
0.8660
1.5708
1.0000
2.0944
2.6180
3.1416
3.6652
4.1888
4.7124
5.2360
5.7596
6.2832
Given y = sin (x)
Table of values
Graph
What if we reversed the ordered pairs … y for x ?

3. Reversed Ordered Pairsx y
0.0000
0.5000
0.5236
0.8660
1.0472
1.0000
1.5708
2.0944
2.6180
3.1416
3.6652
4.1888
4.7124
5.2360
5.7596
6.2832
Problem
This is not a function
Fails the vertical line test
There are multiple (x,y)'s where x = .5
Solution
Limit the range

4. The Inverse Trig Function
We say
Similarly for inverse cosine
The range of cos-1x is limited differently
Note pg 258 for domain, range of other functions

5. Evaluating Inverse Functions
Consider cos-1(-0.5)
We are asking what angle has a cosine value of -0.5
Cosine negative in quadrants 2 and 3
But for cos-1(x) we look only in 1 & 2
Calculator also capable of evaluating inverse trig functions 2 -1

6. Note: newer calculators will have these functions – find in Catalog
Try It Out
Consider these
Note: newer calculators will have these functions – find in Catalog

7. Composition of Trig Functions and Inverses
Recall that in general
f-1(f(x)) = f(f-1(x)) = x
For trig functions this is the same
sin(arcsin(x)) = arcsin(sin(x))
The restriction on the domain and range of the inverse functions must apply
Thus
sin-1(sin(3)) could not be 3
Note calculator results

8. Composition of Trig Functions and Inverses
Try these … with and without calculator

9. Solving Inverse Trig Equations
Given
Strategy
Isolate the sin-1x
Take the sine of both sides of the equation

10. Try it Out
Try this one

11. Assignment
Lesson 3.5
Page 265
Exercises 1 – 65 EOO