1. Trig/Precalc Chapter 5.7 Inverse trig functions
Objectives
Evaluate and graph the inverse sine function
Evaluate and graph the remaining five inverse trig functions
Evaluate and graph the composition of trig functions

2. The basic sine function fails the horizontal line test
The basic sine function fails the horizontal line test. It is not one-to-one so we can’t find an inverse function unless we restrict the domain. Highlight the curve –π/2 < x < π/2 π 2π
π/2
-π/2
y = sin(x)
We switch x and y to get inverse functions So for f(x) = sin-1 x
the domain is [-1, 1] and
range is [-π/2, π/2]
On the interval [-π/2, π/2] for sin x:
the domain is [-π/2, π/2] and the range is [-1, 1]
Therefore

3. Graphing the Inverse
Next we rotate it across the y=x line producing this curve
First we draw the sin curve
When we get rid of all the duplicate numbers we get this curve
This gives us:
Domain : [-1 , 1]
Range:

4. Inverse sine function y = sin-1 x or y = arcsin x
The sine function gives us ratios representing opposite over hypotenuse in all 4 quadrants.
The inverse sine gives us the angle or arc length on the unit circle that has the given ratio.
π/2 1
-π/2
Remember the phrase “arcsine of x is the angle or arc whose sine is x”.

5. Evaluating Inverse Sine
If possible, find the exact value.
arcsin(-1/2) = ____
We need to find the angle in the range
[-π/2, π/2] such that sin y = -1/2
What angle has a sin of ½? _______
What quadrant would it be negative and within the range of arcsin? ____
Therefore the angle would be ______

6. Evaluating Inverse Sine cont.
b. sin-1( ) = ____
We need to find the angle in the range [-π/2, π/2] such that sin y =
What angle has a sin of ? _______
What quadrant would it be positive and within the range of arcsin? ____
Therefore the angle would be ______
c. sin-1(2) = _________ √ 1

7. Graphs of Inverse Trigonometric Functions
The basic idea of the arc function is the same whether it is arcsin, arccos, or arctan

8. Inverse Functions Domains and Ranges
y = arcsin x
Domain: [-1, 1]
Range:
y = arccos x
Domain: [ -1, 1]
y = arctan x
Domain: (-∞, ∞)
y = Arcsin (x)
y = Arccos (x)
y = Arctan (x)

9. Evaluating Inverse Cosine
If possible, find the exact value.
arccos(√(2)/2) = ____
We need to find the angle in the range
[0, π] such that cos y = √(2)/2
What angle has a cos of √(2)/2 ? _______
What quadrant would it be positive and within the range of arccos? ____
Therefore the angle would be ______
b. cos-1(-1) = __
What angle has a cos of -1 ? _______

10. Warnings and Cautions!
Inverse trig functions are equal to the arc trig function. Ex: sin-1 θ = arcsin θ
Inverse trig functions are NOT equal to the reciprocal of the trig function.
Ex: sin-1 θ ≠ 1/sin θ
There are NO calculator keys for: sec-1 x, csc-1 x, or cot- 1 x
And csc-1 x ≠ 1/csc x
sec-1 x ≠ 1/sec x
cot-1 x ≠ 1/cot x

11. Evaluating Inverse functions with calculators ([E] 25 & 34)
If possible, approximate
to 2 decimal places.
19. arccos(0.28) = ____
22. arctan(15) = _____
26. cos-1(0.26) = ____
34. tan-1(-95/7) = ____
Use radian mode unless degrees are asked for.

12. You Try! Evaluate:
csc[arccos(-2/3)] (Hint: Draw a triangle)