1. Keeper 11 Inverses of the Trigonometric Functions
Find values of the inverse trigonometric functions.
Simplify expressions such as sin (sin -1 x) and sin -1 (sin x).
Simplify expressions involving composition such as sin (cos –1 1/2) without using a calculator.
Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.
Copyright © 2009 Pearson Education, Inc.

2. Copyright © 2009 Pearson Education, Inc.
Inverse Sine Function
The graphs of an equation and its inverse are reflections of each other across the line y = x.
However, the inverse is not a function as it is drawn.
Copyright © 2009 Pearson Education, Inc.

3. Copyright © 2009 Pearson Education, Inc.
Inverse Sine Function
We must restrict the domain of the inverse sine function.
It is fairly standard to restrict it as shown here.
The domain is [–1, 1].
The range is [–π/2, π/2].
Copyright © 2009 Pearson Education, Inc.

4. Inverse Cosine Function
The graphs of an equation and its inverse are reflections of each other across the line y = x.
However, the inverse is not a function as it is drawn.
Copyright © 2009 Pearson Education, Inc.

5. Inverse Cosine Function
We must restrict the domain of the inverse cosine function.
It is fairly standard to restrict it as shown here.
The domain is [–1, 1].
The range is [0, π].
Copyright © 2009 Pearson Education, Inc.

6. Inverse Tangent Function
The graphs of an equation and its inverse are reflections of each other across the line y = x.
However, the inverse is not a function as it is drawn.
Copyright © 2009 Pearson Education, Inc.

7. Copyright © 2009 Pearson Education, Inc.
Inverse Sine Function
We must restrict the domain of the inverse tangent function.
It is fairly standard to restrict it as shown here.
The domain is (–∞, ∞).
The range is (–π/2, π/2).
Copyright © 2009 Pearson Education, Inc.

8. Inverse Trigonometric Functions
Function Domain Range
Copyright © 2009 Pearson Education, Inc.

9. Graphs of the Inverse Trigonometric Functions
Copyright © 2009 Pearson Education, Inc.

10. Graphs of the Inverse Trigonometric Functions
Copyright © 2009 Pearson Education, Inc.

11. Copyright © 2009 Pearson Education, Inc.
Example
Find each of the following function values.
Solution:
Find  such that sin  =
In the restricted range [–π/2, π/2], the only number with sine of is π/4.
Copyright © 2009 Pearson Education, Inc.

12. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Find  such that cos  = –1/2.
In the restricted range [0, π], the only number with cosine of –1/2 is 2π/3.
Copyright © 2009 Pearson Education, Inc.

13. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Find  such that tan  =
In the restricted range (–π/2, π/2), the only number with tangent of is –π/6.
Copyright © 2009 Pearson Education, Inc.

14. Copyright © 2009 Pearson Education, Inc.
Example
Approximate the following function value in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree.
Solution:
Press the following keys (radian mode):
Readout:
Rounded:
Change to degree mode and press the same keys:
Readout:
Rounded: 105.5º
Copyright © 2009 Pearson Education, Inc.

15. Composition of Trigonometric Functions
for all x in the domain of sin–1
for all x in the domain of cos–1
for all x in the domain of tan–1
Copyright © 2009 Pearson Education, Inc.

16. Copyright © 2009 Pearson Education, Inc.
Example
Simplify each of the following.
Solution:
a) Since is in the domain, [–1, 1], it follows that
b) Since 1.8 is not in the domain, [–1, 1], we cannot evaluate the expression. There is no number with sine of 1.8. So, sin (sin–1 1.8) does not exist.
Copyright © 2009 Pearson Education, Inc.

17. Copyright © 2009 Pearson Education, Inc.
Special Cases
for all x in the range of sin–1
for all x in the range of cos–1
for all x in the range of tan–1
Copyright © 2009 Pearson Education, Inc.

18. Copyright © 2009 Pearson Education, Inc.
Example
Simplify each of the following.
Solution:
a) Since π/6 is in the range, (–π/2, π/2), it follows that
b) Since 3π/4 is not in the range, [–π/2, π/2], we cannot apply sin–1(sin x) = x.
Copyright © 2009 Pearson Education, Inc.

19. Copyright © 2009 Pearson Education, Inc.
Example
Find
Solution:
cot–1 is defined in (0, π), so consider quadrants I and II. Draw right triangles with legs x and 2, so cot  = x/2.
Copyright © 2009 Pearson Education, Inc.