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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
1
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
y = sin(x)
-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
π/2
π
2π
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]
2
10
Graphing the Inverse
First we draw the sin curve
Next we rotate it across the
y=x line producing this curve
5
6
4
5
-6
-4
-2
2
-5
4
-10
6
2
10
-2
This gives us:
Domain : [-1 , 1]
-4
-5
When we get rid of all the
duplicate numbers we get
this curve
-6


Range:
2,  2 
3
Inverse sine function
-1
y = sin x or y = arcsin x
• The sine function gives us ratios
representing opposite over
hypotenuse in all 4 quadrants.
4
2
π/2
• The inverse sine gives us
-5 the angle
or arc length on the unit circle that
has the given ratio.
1
5
-π/2
-2
Remember the phrase “arcsine of x is the
angle or arc whose sine is x”.
-4
4
Evaluating Inverse Sine
If possible, find the exact value.
a. 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 ______
5
Evaluating Inverse Sine cont.
b. sin-1( 3) = ____
2
We need to find the angle in the range [-π/2, π/2] such that sin y
3
=
2
3
2
√3
2
What angle has a sin of
? _______
1
What quadrant would it be positive and within the range of
arcsin? ____
Therefore the angle would be ______
c. sin-1(2) = _________
6
Graphs of Inverse Trigonometric
Functions
The basic idea of the arc function is the same
whether it is arcsin, arccos, or arctan
7
Inverse Functions Domains and Ranges
• y = arcsin x
y = Arcsin (x)
• Domain: [-1, 1]
• Range:    
 , 
 2 2
• y = arccos x
• Domain: [ -1, 1]
• Range:
0,  
y = Arccos (x)
• y = arctan x
• Domain: (-∞, ∞)
• Range:    
 , 
y = Arctan (x)
 2 2
8
Evaluating Inverse Cosine
If possible, find the exact value.
a.
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 ? _______
9
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 cot1x
And csc-1 x ≠ 1/csc x
sec-1 x ≠ 1/sec x
cot-1 x ≠ 1/cot x
10
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.
11
You Try!
Evaluate:

3
arcsin  

2



 3 
tan  arccos    
 5 

3 

arcsin  sin

2


arccos  tan 2 
csc[arccos(-2/3)] (Hint: Draw a triangle)