Inverse Trigonometric Functions
section 4.7
* Recall that a function may only have an inverse if it is one-to-one (i.e. it must pass
the horizontal line test). The sine function does not pass this test, so it does not
have an inverse function. However, if we restrict its domain in a certain
way, then it will pass the test and thus will have an inverse.
y = sin x
(restricted domain)
Notice that the sine function takes on the full range of y-values [-1, 1] over the
restricted domain interval of [-
2,
2 ] . So, on that restricted domain,
sin x does have a unique inverse called theinverse sine function. It can be written
either as y = arcsin x or y = sin -1 x. Be careful that you realize sin -1 x means
the inverse sine function and not 1/sinx.
Definition of Inverse Sine Function
y = arcsin x if and only if siny = x
Domain of sin x: -
/2
Range of sin x: -1
y 1
x
Domain of arcsin x: -1
/ 2.
Range of arcsin x: -
x
/2
1
y
/2.
* Note that the domain and range of the sin and sin -1 functions switch places with
one another. Each x-value in one function equals the y-value in the other and
vice versa.
problem #1 - If possible, find the exact value of the following:
-
a. arcsin
1
2
3
2
b. sin -1
c. sin -1 (2)
The arcsine function can be graphed by plotting several points:
y
-
x = sin y
-
2
- 6
4
- 2
2
-1
0
1
- 2
4
2
2
6
1
2
0
(1,
2
1
(
y = arcsin x
( 12
(0, 0)
-1
(- 12 , - 6 )
2
(- 2 , - 4 )
2
1
2
2
,
2
4
)
)
,6)
1
-1
(-1 , - 2 )
-
2
* Remember that the values of the inverse sine function are always given in radians.
The cosine function also does not pass the horizontal line test and thus has no inverse.
But, we can restrict its domain, as with the sine function, to obtain an interval in which
it will have an inverse.
Domain of cos x:
x
0
Range of cos x: -1
y
Domain of arccos x: -1
1
Range of arccos x:
0
x
1
y
The cosine function takes on the full range of y-values [-1, 1] over the restricted
domain interval of [0 ,
] . The inverse cosine function can be written as either
y = arccos x or y = cos -1x.
* Continued on next page *
section 4.7 (continued)
The arccosine function can be graphed by plotting several points:
y
0
x = cos y
1
6
3
2
(-1,
(-
,5
3
2
3
1
2
2
3
1
- 2
2
0
)
5
6
3
2
-
-1
y = arccos x
)
6
(- 1 , 2
2
3
)
(0, 2 )
(
1 ,
)
2 3
( 3
2
,
6
)
(1, 0)
0
-1
1
The inverse tangent function can be created in the same way as the inverse sine and
cosine functions. Its restricted domain is [2,
2].
Domain of tan x: -
/2
Range of tan x: -
x
Domain of arctan x: -
/2.
y
Range of arctan x: -
x
/2
y
/2.
2
(1,
y = arctan x
(-1,-
6
4
)
)
-
2
problem #2 - Find the exact values of:
a. arccos
2
2
b. cos -1 (-1)
c. arctan 0
d. tan -1 (-1)
problem #3 - Use a calculator to approximate the following values, if possible.
b. sin -1 0.2447
a. arctan (- 8.45)
c. arccos 2
Compositions of Functions & Inverse Properties
If -1 x
1 and
-
y
2
2
, then
sin (arcsin x) = x and arcsin (sin y) = y
If -1 x 1 and
, then
0 y
cos (arccos x) = x and arccos (cos y) = y
If x is a real number and
-
y
2
2
, then
tan (arctan x) = x and arctan (tan y) = y
problem #4 - If possible, find the exact values.
b. arcsin sin 5
3
a. tan [arctan (-5)]
c. cos (cos -1
)
problem #5 - Find the exact values. Hint: draw a right triangle.
a. tan arccos
2
3
b. cos
arcsin
-
3
5
* One important role of inverse trigonometric functions is in solving trigonometric
equations in which the angle is the unknown quantity. They are also very important
in the study of calculus. An example of this is found in the problem below.
problem #6 - Write each of the following as an algebraic expression inx.
a. sin (arccos 3x),
b. cot (arccos 3x),
0
0
x
x
1
3
1
3