6.1 Inverse Circular Functions
A relation is any set of ordered pairs. A function is a relation in which no two different ordered
pairs have the same first coordinate. We define the inverse of a relation R to be
, | ,
.
1. Determine the inverse of each function. Is the inverse also a function?
a)
2, 8 , 1, 1 , 0,0 , 1,1 , 2,8
b)
2,4 , 1,1 , 0,0 , 1,1 , 2,4
Solution:
a)
8, 2 , 1, 1 , 0,0 , 1,1 , 8,2
is also a function since no two different ordered pairs have the same first coordinate.
b)
4, 2 , 1, 1 , 0,0 , 1,1 , 4,2
is not a function since 4, 2
4,2 are tow different ordered pairs with the same first
coordinate.
Functions whose inverse are also functions are called one-to-one functions. In a one-to-one
function, for each value of x there corresponds exactly one value of y and for each value of y
there corresponds exactly one value of x. From the previous example, 1(a) is a one-to-one
function, but 1(b) is not a one-to-one function since its inverse is not a function.
2. Graph
| |. Is f one-to-one?
f is not one-to-one, since the y-value 2 corresponds to more than one x-value namely -2 and 2.
Horizontal Line Test: A function f is one-to-one if and only if every horizontal line intersects
the graph of f in at most one point.
sin .
Recall the graph of
sin is not one-to-one. In order to define its
Applying the horizontal line test, we see
inverse we need to restrict its domain to
Inverse Sine Function
sin
or
sin means that
,
so that its inverse will also be a function.
sin for
.
3. Find the exact value.
b) sin
a) sin
Solution: a. sin
c. sin
c) sin
√
d) sin
√
e) sin
b. sin
√
d. sin
√
e. sin
1
1
cos
Recall the graph of
Applying the horizontal line test, we see
inverse we need to restrict its domain to 0,
cos is not one-to-one. In order to define its
so that its inverse will also be a function.
Inverse Cosine Function
cos
or
cos means that
cos , for 0
4. Find the exact value:
a) cos
√
Solution: a. cos
d. cos
b) cos
√
√
√
c) cos
√
b. cos
e. cos
1
d) cos
c. cos
e) cos
√
1
Restricting the domain of
tan to
,
yields a one-to-one function which has an
inverse function.
Inverse Tangent Function
tan
or
tan means that
tan ,
5. Find the exact value:
a) tan
√3
b) tan
Solution: a. tan
√3
√3
d. tan
c) tan
b. tan
1
√
d) tan
√3
e. tan
1
e) tan
√
c. tan
√
Find the exact value of each expression:
6. tan
cos
(3,y)
cos
Solution:
4
16
16
√7
3
9
tan
cos
9
3
4
means cos
7
tan
√7
3
r=4
√
7. sec sin
Solution:
sin
since sin
0.
means sin
. The angle
must be in the 4th quadrant
r=5
(x,-1)
1
1
5
25
24
2√6
√24
1
5
sec sin
8. cos sin
5
sec
5√6
12
2√6
cos
sin
Solution:
2√6
means sin
cos
and
means cos
(x,3)
r=5
r = 13
For angle A
3
9
For angle B
5
25
16
5
25
4
√16
cos sin
cos sin
3
5
3
5
cos
cos
5
13
5
13
cos
20
65
(5,y)
cos cos
36
65
16
65
4
13
169
144
√144
sin sin
4 5
·
5 13
12
3 12
·
5 13
12