1. Inverse Trigonometric Functions
Trigonometry
MATH 103
S. Rook

2. 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

3. Review of Inverse Functions

4. 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

5. 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

6. Inverse Sine Function

7. 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

8. 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

9. 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

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

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

12. Inverse Cosine Function

13. 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

14. 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

15. 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

16. 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)

17. Inverse Tangent Function

18. 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

19. 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

20. 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

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

22. Inverse Trigonometric Functions and Right Triangles

23. 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

24. 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

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

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

27. 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)