Trig/Precalc
Chapter 4.7 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
y = sin-1 x or y = arcsin x
 The sine function gives us
ratios representing opposite
over hypotenuse in all 4
quadrants.
4
2
π/2
1
-5
 The inverse sine gives us the
angle or arc length on the unit
circle that has the given ratio.
Remember the phrase “arcsine of x is the
angle or arc whose sine is x”.
-π/2
-2
-4
4
Evaluating Inverse Sine
If possible, find the exact value.

6
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? _______
6
What quadrant would it be negative and within
the range of arcsin? ____
IV


Therefore the angle would be ______
6
5
Evaluating Inverse Sine cont.
b.
3
-1
sin ( 2 )

= ____
3
We need to find the angle in the range [-π/2, π/2] such that
sin y = 3
√3
2
2

3
What angle has a sin of
? _______
1
What quadrant would it be positive and within the range of
I
arcsin? ____

Therefore the angle would be ______
3
3
2
No Solution
c. sin-1(2) = _________
Sin domain is [-1, 1], therefore No solution
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
 Domain: [-1, 1]
 Range:    ,  
y = Arcsin (x)
 2 2 
 y = arccos x
 Domain: [ -1, 1]
 Range:  0,  
 y = arctan x
 Domain: (-∞, ∞)
 
 Range:   , 
 2 2
y = Arccos (x)
y = Arctan (x)
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 cot-1 x
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
Guided practice
Example of [E] 28 & 30
Use an inverse trig function “θ as a function of x”
means to write an equation
to write θ as a function of x. of the form θ equal to an
expression with x in it.
28.
Cos θ = 4/x so
x
θ = cos-1(4/x) where x > 0

4
30.

10
tan θ = (x – 1)/(x2 – 1)
x  1 θ = tan-1(x – 1)/(x2 – 1)
where x – 1 > 0 , x > 1
12
Composition of trig functions
Find the exact value, sketch a triangle.
cos(tan-1 (2)) = _____
This means tan θ = 2 so…
draw the triangle
Label the adjacent and opposite sides
√
5
2
θ
1
Find the hypo. using Pyth. Theorem
2 5
So the cos  
5
13
Example
Write an algebraic expression that is equivalent to
the given expression.
cos(arctan(1/x))
1) Draw and label the triangle
x2 1
1
---(let u be the unknown angle)
2) Use the Pyth. Theo. to compute the hypo
3) Find the cot of u
cos u 
u
x
x x2 1

2
2
x
1
x 1
x
14
You Try!
Evaluate:

3
arcsin  

2


 3
3 

arcsin  sin

2


 2

 3 
tan  arccos     -4/3
 5 

arccos  tan 2  0 rad.
csc[arccos(-2/3)] (Hint: Draw a triangle)
3 5 5
Rewrite as an algebraic expression:
v2  1
v2  1
A
L
E
K
S
Word problem involving sin or cos function:
P type 1
An object moves in simple harmonic motion with amplitude 12 cm
and period 0.1 seconds. At time t = 0 seconds , its displacement
d from rest is 12 in a negative direction, and initially it moves in
a negative direction.
Give the equation modeling the displacement d as a function of
time t.
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A
L
E
K
S
Word problem involving sin or cos function:
P type 2
The depth of the water in a bay varies throughout the day with the tides.
Suppose that we can model the depth of the water with the following
function.
h(t) = 13 + 6.5 sin 0.25t
In this equation, h(t) is the depth of the water in feet, and t is the time in
hours.
Find the following. If necessary, round to the nearest hundredth.
Frequency of h:
cycles per hour
Period of h:
hours
Minimum depth of the water: feet
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