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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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