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Inverse
Trigonometric
Functions
4.7
The Inverse Sine Function
The inverse sine function, denoted by sin-1, is the inverse of the restricted sine
function y = sin x, -  /2 < x <  / 2. Thus,
y = sin-1 x means sin y = x,
where -  /2 < y <  /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the
inverse sine at x.”
y
y = sin x
-  /2 < x < /2
1
-  /2
 /2
-1
x
Domain: [-  /2,  /2]
Range: [-1, 1]
Finding Exact Values of
-1
sin x
• Let  = sin-1 x.
• Rewrite step 1 as sin  = x.
• Use the exact values in the table to find the
value of  in [-/2 , /2] that satisfies sin 
= x.
Example
• Find the exact value of sin-1(1/2)
1 1
  sin
2
1
sin  
2
 1
sin 
6 2


6
Example
• Find the exact value of sin-1(-1/2)
1
  sin
2
1
sin   
2

1
sin

6
2
1
 

6
The Inverse Cosine Function
The inverse cosine function,denoted by cos-1,
is the inverse of the restricted cosine
function
y = cos x, 0 < x < . Thus,
y = cos-1 x means cos y = x,
where 0 < y <  and –1 < x < 1.
Text Example
Find the exact value of cos-1 (-3 /2)
 3
  cos
2
 3
cos  
2
5  3
cos

6
2
5

6
1
Text Example
Find the exact value of cos-1 (2 /2)
  cos
1
2
2
2
cos  
2

2
cos 
4
2


4
The Inverse Tangent Function
The inverse tangent function, denoted by
tan-1, is the inverse of the restricted tangent
function
y = tan x, -/2 < x < /2. Thus,
y = tan-1 x means tan y = x,
where -  /2 < y <  /2 and –  < x < .
Text Example
Find the exact value of tan-1 (-1)
  tan 1  1
tan   1

tan
4


4
 1
Text Example
Find the exact value of tan-1 (3)
  tan 1 3
tan   3
tan


3

3
 3
Inverse Properties
The Sine Function and Its Inverse
sin (sin-1 x) = x for every x in the interval [-1, 1].
sin-1(sin x) = x for every x in the interval [-/2,/2].
The Cosine Function and Its Inverse
cos (cos-1 x) = x for every x in the interval [-1, 1].
cos-1(cos x) = x for every x in the interval [0, ].
The Tangent Function and Its Inverse
tan (tan-1 x) = x for every real number x
tan-1(tan x) = x for every x in the interval (-/2,/2).
Example
cos cos 1 0.3
3 
sin 1  sin

2 

cos cos 1 4.6
Example
5
sin  tan 1 
12 

Example
1
cot  sin 1  
3

Example
cos sin 1 x
Using you Calculator
Find the angle in radians to the nearest thousandth. Then find the
angle in degree.
sin 1
cos
1
1
4
1
3
tan 1 10.25
tan 1 (43.7)
Example
• The following formula gives the viewing
angle θ, in radians, for a camera whose lens
is x millimeters wide. Find the viewing
angle in radians and degrees for a 28
millimeter lens.
  2 tan 1
21.634
x
  2 tan 1
21.634
28
  1.3157 radians or 75.4