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CH. 4 – TRIGONOMETRIC
FUNCTIONS
4.7 – Inverse Trig Functions
INVERSE SINE FUNCTION

Does y = sinx have an inverse?
No, because it fails the horizontal line test!
 However, let’s restrict the domain of y = sinx to [-π/2, π/2]
so that…
 The function is one-to-one
 y = sinx takes on its full range of values
 …then we have the inverse function, y = sin-1 x
 It’s also called y = arcsin x
 Domain: [-1, 1]
 Range: [-π/2, π/2]

y = sin x if and only if sin-1 y = x
 Remember: y = sin-1 x outputs an angle!

1
sin
.
2
1
FIND THE EXACT VALUE OF
1.

3
Mr. Weinmann, show the class how
to solve this problems!

6
2.
3.
5
6
4.
5.


3

4
0%
0%
0%
0%
0%
arcsin  1 .
FIND THE EXACT VALUE OF
1.


2
2.
3.
3
2

4.
5.
3

2

2
0%
0%
0%
0%
0%
GRAPHING Y = SIN-1 X
Domain: [-1, 1], Range: [-π/2, π/2]
 Make a table recalling points from the unit circle
 Remember, it should look like a piece of an inverted
sine function

Y values are in radians!
 The graph should be one-to-one
 No arrows on the end of the graph this time!

X
Y
-1
-π/2
-½
-π/6
0
0
½
π/6
1
π/2
by ½’s
by π/6’s
INVERSE COSINE FUNCTION

Like the sine function, the cosine function must be
restricted to have an inverse
We restrict the domain of y = cosx to [0, π] so that…
 The function is one-to-one
 y = cosx takes on its full range of values
 …then we have the inverse function, y = cos-1 x
 It’s also called y = arccos x
 Domain: [-1, 1]
 Range: [0, π]


y = cos x if and only if cosy = x
 3
cos 

 2 
1
FIND THE EXACT VALUE OF
1.

4

2
2.
3.


6
4.
5.
.

3

6
0%
0%
0%
0%
0%

2
cos  
 .
 2 
1
FIND THE EXACT VALUE OF
1.

4
3
4
2.
3.


4
4.
5.
5

4
5
4
0%
0%
0%
0%
0%
INVERSE TANGENT FUNCTION


Since we don’t cover tangent graphs, just believe me when I say
that for y = tan-1 x…
 Domain: [-∞, ∞] or all real numbers
 Range: [-π/2, π/2]
Ex: Find tan-1 ( 3 ) in radians.

This will be tough to figure out by hand, so just use your calculator!
If we do this problem with our calculator in radians, the answer
probably will be a decimal…
…so do the problem in degrees and convert to radians!

You should get 60°, or π/3!


tan 1  0 
FIND THE EXACT VALUE OF
1.
0

4
2.
3.

2

4.
5.
.

3

2
0%
0%
0%
0%
0%
cos  0 
FIND THE EXACT VALUE OF
1.
1
2.
0
3.

2
4.
undefined
5.

4
0%
0%
0%
.
0%
0%

Ex: Find sin-1 (sin(


5
)).
6
Inverse sine and sine will cancel, but we have to make sure the angle is in the
range of inverse sine.
Since 5 is not in [-π/2, π/2], find an angle with that sine that is in the
range…6


…so we get
6
3

2
Ex: Find tan(arccos( 3 )).
5
θ
2




Draw a right triangle to help you!
This question is asking, “what is the tangent if the cosine is 2/3?”
Complete the right triangle…
5
…to get
2
FIND THE EXACT VALUE OF
1.
4.
5.

) .

1
3
4
2.
3.
 3
sin (sin 
 4
1
2
2

4
undefined
0%
0%
0%
0%
0%
3
tan(sin  )
4
1
FIND THE EXACT VALUE OF
1.
0
7
4
2.
3.
4.
5.
.
3 7
7
3
4
undefined
0%
0%
0%
0%
0%