1. Section 4.6 Inverse Trigonometric fuctions
Objectives:
-Evaluate and Graph inverse trig functions
-Find compositions of trig functions

2. Inverse SINE Is the sine function one-to-one?
No! Does not pass HLT.
If we restrict the domain of the sine function to the interval [− 𝜋 2 , 𝜋 2 ], the function IS one-to-one.
The inverse equation is y = sin -1x and is graphed by reflecting the restricted y = sinx in the line y = x.

3. Inverse SINE
Notice that the domain of the inverse is [-1, 1] and its range is [− 𝜋 2 , 𝜋 2 ].
Because angles and arcs given on the unit circle have equivalent radian measures, the inverse sine function is sometimes referred to as the arcsine function y = arcsin x.

4. Inverse SINE
sin -1 x or y = arcsin x can be interpreted as the angle (or arc) between − 𝜋 2 𝑎𝑛𝑑 𝜋 2 with a sine of x.
For example, sin is the angle with a sine of 0.5.
(30 degrees)
Recall that sin t is the y-coordinate of the point on the unit circle that corresponds to the angle or arc length, t.
Because the range of the inverse sine function are restricted, the possible angle measures of the inverse sine function are located on the right half of the unit circle

5. Example 1: Find the exact value, if it exists.
A) sin −1 2 2 B) arcsin − 3 2 C) sin −1 −2 D) sin −1 0

6. Inverse COSINE
To make the cosine function one-to-one, the domain must be restricted to [0, ].
The inverse cosine function is y = cos -1x or arccosine function y = arccos x.
The graph of y = cos -1 x is found by reflecting the graph of the restricted y = cosx in the line y = x.

7. Inverse COSINE
Recall that cos t is the x-coordinate of the point on the unit circle that corresponds to the angle or arc length, t.
Because the range of y = cos -1x is restricted to [0, ] , the possible angle measures of the inverse cosine function are located on the upper half of the unit circle

8. Example 2: Find the exact value, it exists.
A) cos −1 1 B) arccos − 3 2 C) cos −1 3 D) cos −1 −1

9. Inverse TANGENT
To make the tangent function one-to-one, the domain must be restricted to (− 𝜋 2 , 𝜋 2 ).
The inverse tangent function is y = tan-1x or arctangent function y = arctan x.
The graph of y = tan-1 x is found by reflecting the graph of the restricted y = tanx in the line y = x.
Unlike sine and cosine, the domain of the
inverse tangent function is (-, )

10. Inverse TANGENT On the unit circle, tant = sin 𝑥 cos 𝑥 or tant = 𝑦 𝑥
The values of y = tan-1x will be located on the right half of the unit circle, not including − 𝜋 2 and 𝜋 2 because the tangent function is undefined at those points.

11. Example 3: Find the exact value, it exists.
A) tan −1 3 3 B) arctan 1 C) arctan (− 3 )

12. SUMMARY

13. Graphing Inverse Trig Functions
Rewrite the function in one of the following forms:
sin y = x
cos y = x
tan y = x
Make a table of values, assigning radians from the restricted range to y.
Plot the points and connect them with a smooth curve.

14. Example 4: Sketch the graph
A) y = arctan 𝑥 2 x y

15. Example 4: Sketch the graph
B) y = sin −1 2𝑥 x y

16. Example 4: Sketch the graph
C) y = arccos 𝑥 4 x y

17. Example 5: Application
A) In a movie theater, a 32-foot-tall screen is located 8 feet above ground. Write a function modeling the viewing angle θ for a person in the theater whose eye-level when sitting is 6 feet above ground.

18. Example 5:
B) Determine the distance that corresponds to the maximum viewing angle.

19. Composition of Trig Functions
In Lesson 1.7, you learned that if x is in the domain of f(x) and f -1(x) then
f [f -1(x)] = x and f -1[f(x)] = x
Because the domains of the trig functions are restricted to obtain the inverse trig function, the properties do not apply for all values of x.
For example, while sin x is defined for all x, the domain of sin-1 x is [-1,1].
Therefore, sin(sin-1 x) = x is only true when -1 ≤ x ≤ 1.
A different restriction applies for the composition of sin-1(sinx) because the domain of sin x is [− 𝜋 2 , 𝜋 2 ].
Therefore, sin-1(sinx) = x is only true when − 𝝅 𝟐 ≤𝒙≤ 𝝅 𝟐

20. SUMMARY of Composition Domain Restrictions

21. Example 6: Find the exact value, if it exists
A) sin⁡ arcsin 1 2 B) cos −1 cos 5𝜋 2 C) arctan⁡ tan − 5𝜋 2 D) arcsin sin 7𝜋 6

22. Example 7: Find the exact value
A) sin⁡ cos −1 4 5 B) cos arcsin 8 17 C) tan arccos − 5 13

23. Reciprocal Inverses arccosecant (sec -1) arcsecant (csc -1)
arccotangent (cot -1)
EXTRA EXAMPLE: Find the exact value.
A) csc cot − B) tan csc −
C) sec cos −1 − D) cot sec −1 − 25 7

24. Example 8:
A) Write cot (arccos x) as an algebraic expression of x that does not involve trigonometric functions.

25. Example 8:
B) Write cos(arctan x) as an algebraic expression of x that does not involve trigonometric functions.

26. Take A Picture For HW