Download Inverse Trigonometric Functions — 7.6

Math 4 Pre-Calculus
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Inverse Trigonometric Functions — 7.6
Inverse Sine Function
The inverse sine function, denoted by sin −1 x or arcsin x , is defined by
y = sin −1 x
y = arcsin x
for −1 ≤ x ≤ 1 and −
π
2
≤ y ≤
if and only if
if and only if
π
2
x = sin y
x = sin y
. This limits the range to angles in Quadrants I and IV.
Graph y = sin −1 x = arcsin x .
Domain:
1.
Range:
Symmetry:
Properties of Inverse Sine
(
)
sin sin −1 x = sin ( arcsin x ) = x
if
−1 ≤ x ≤ 1
sin −1 ( sin y ) = arcsin ( sin y ) = y
if
−
2.
1

sin  sin −1 
2

3.
2π 

arcsin  sin
3 

π
2
≤ y ≤
π
2
Inverse Cosine Function
The inverse cosine function, denoted by cos−1 x or arccos x , is defined by
y = cos −1 x
y = arc cos x
if and only if
if and only if
x = cos y
x = cos y
for −1 ≤ x ≤ 1 and 0 ≤ y ≤ π . This limits the range to angles in Quadrants I and II.
Graph y = cos −1 x = arccos x .
Domain:
4.
Range:
Properties of Inverse Cosine
(
)
cos cos −1 x = cos ( arccos x ) = x
if
−1 ≤ x ≤ 1
cos −1 ( cos y ) = arccos ( cos y ) = y
if
0 ≤ y ≤ π
5.
cos cos −1 ( −0.5 ) 


6.
arccos ( cos π )
Inverse Tangent Function
The inverse tangent function, denoted by tan −1 x or arctan x , is defined by
y = tan −1 x
y = arctan x
for any real number x and −
π
2
if and only if
if and only if
< y <
π
2
x = tan y
x = tan y
. This limits the range to angles in Quadrants I and IV.
Graph y = tan −1 x = arctan x .
Domain:
7.
Range:
Horizontal Asymptotes:
Properties of Inverse Tangent
(
)
tan tan −1 x = tan ( arctan x ) = x
tan −1 ( tan y ) = arctan ( tan y ) = y
(
8.
tan tan −1 2500
9.
arctan ( tan π )
)
for every real number x
if
−
π
2
< y <
π
2
Find the exact value of the expression whenever it is defined.
( )
10.
s i n−1 −
11.
t a n c o s− 1 0
12.
c o t s i n − 1 − 52
13.
cos
3
2
(
)
(
( ))
(
1
2
arctan
( 185 ) )
Write the expression as an algebraic expression in x for x > 0 .
14.
t an (arcco s x)
15.

c o t  s i n− 1


x2 − 9 


x

The given equation has the form y = f ( x ) a) Find the domain of f b) Find the range of f c) Solve for x in terms of y .
16.
y = 2 s i n − 1 ( 3x − 4 )
Solve the equation for x in terms of y if x is restricted to the given interval.
17.
y = 2 + 3sin x
 π π
− 2 , 2 


Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the
solutions to four decimal places.
 π π
18.
6 s i n 2x − 8 c o s x + 9 s i n x − 6 = 0
− , 
 2 2