Section 6.1
Inverse Circular Functions
Recall from College Algebra that not every function has an inverse function. In order for a function to have an
inverse function, it must be one-to-one.
Properties of Inverse Functions
The inverse of a function f is denoted by f 1 , read “f-inverse”.
Since x and y values switch roles between inverses, if (x, y) is on𝑓(𝑥), then (y, x) is on 𝑓 (𝑥 ).
The domain of f is the range of f 1 and the range of f is the domain of f 1 .
The graphs of f and f 1 are reflections of each other across the line 𝑦 = 𝑥.
A function is one-to-one, or has an inverse function, if and only if NO horizontal line crosses the graph
more than once.
Recall the function, 𝑓(𝑥 ) = 𝑥 .
What is its inverse function? 𝑓
How could we restrict the domain
to make it one-to-one?
What is its domain?
What is its range?
What is its range?
(𝑥 ) =
Inverse Sine Function
ORIGINAL FUNCTION
INVERSE FUNCTION
𝒚 = 𝐬𝐢𝐧
𝒚 = 𝐬𝐢𝐧 𝒙
𝟏
𝒙 OR
𝒚 = 𝐚𝐫𝐜𝐬𝐢𝐧 𝒙
**y is the angle in the interval
Restricted domain: Input
− ,
whose sine value is 𝑥. **
𝝅 𝝅
Angles from − 𝟐 , 𝟐
Domain: Input
Sine values from[−1,1]
Range: Output
Range: Output
Sine values from[−1,1]
Angles from − ,
Example 2: Find the exact value of y in each equation.
a)
𝑦 = arcsin
√
b) 𝑦 = sin
−
Inverse Cosine Function
ORIGINAL FUNCTION
𝒚 = 𝐜𝐨𝐬 𝒙
INVERSE FUNCTION
𝒚 = 𝐜𝐨𝐬
𝟏
𝒙 OR
𝒚 = 𝐚𝐫𝐜𝐜𝐨𝐬 𝒙
**y is angle in the interval [𝟎, 𝝅]whose cosine value is 𝑥. **
Restricted domain: Input
Domain: Input
Angles from [𝟎, 𝝅]
Cosine values
from[−1,1]
Range: Output
Coine values from[−1,1]
Range: Output
Angles from [𝟎, 𝝅]
Example 2: Find the exact value of y in each equation.
a)
𝑦 = arccos
√
b) 𝑦 = cos
−
Inverse Tangent Function
ORIGINAL FUNCTION
𝒚 = 𝐭𝐚𝐧 𝒙
INVERSE FUNCTION
𝒚 = 𝐭𝐚𝐧
𝟏
𝒙 OR
𝒚 = 𝐚𝐫𝐜𝐭𝐚𝐧 𝒙
**y is the angle in the interval
Restricted domain: Input
Angles from − ,
− ,
whose tangent value is 𝑥. **
Domain: Input
Tangent values from(−∞, ∞)
Range: Output
Tangent values from
(−∞, ∞)
Range: Output
Angles from − ,
Example 3: Find the exact value of y in each equation.
a)
𝑦 = arctan 1
b) 𝑦 = tan
−√3
Inverse Functions for Cotangent, Secant, and Cosecant
𝑥 can be evaluated by
cot
To evaluate these inverse functions, we can use the reciprocal relationship.
sec
𝑥
𝐜𝐨𝐬
can be evaluated by
𝟏𝟏
𝒙
csc
𝐭𝐚𝐧
𝟏𝟏
𝒙
if x > 0
𝐭𝐚𝐧
𝟏𝟏
+
𝒙
𝟏𝟖𝟎° if x < 0
𝑥 can be evaluated by
𝐬𝐢𝐧
𝟏𝟏
𝒙
*NOTE: cot
0=
Example 4: For each expression, find the exact measure, in degrees, if it exists. Do not use a calculator.
a)
arccsc −
√
b) arcsec(−
√
)
c) cot
−
√
Example 5: Use a calculator to give each value in decimal degrees. Round to the nearest 2 decimal places.
a) 𝜃 = arccos
(−0.13348122)
b) 𝜃 = csc
1.9422833
Example 6: Use a calculator to give each real number value. (Radians) Round to the nearest 6 decimal places.
a) 𝑦 = sin (−0.13348122)
b) 𝑦 = arcsec(−1.2671684)
Summary of the domain and ranges of the Inverse Functions
Inverse Function
Domain
RANGE
Interval in Degrees
Quadrants
[−𝟗𝟎°, 𝟗𝟎°]
I and IV
𝝅
𝝅
, 𝟎) ∪ (𝟎,
𝟐
𝟐
[−𝟗𝟎°, 𝟎°) ∪ (𝟎°,𝟗𝟎°]
I and IV
𝝅 𝝅
,
𝟐 𝟐
(−𝟗𝟎°, 𝟗𝟎°)
I and IV
Interval in Radians
𝝅 𝝅
− ,
𝟐 𝟐
𝑦 = sin
𝑥
[−1,1]
𝑦 = csc
𝑥
(−∞, −1] ∪ [1, ∞)
𝑦 = tan
𝑥
(−∞, ∞)
−
𝑦 = cot
𝑥
(−∞, ∞)
(𝟎, 𝝅)
(𝟎°, 𝟏𝟖𝟎°)
I and II
𝑦 = cos
𝑥
[−1,1]
[𝟎, 𝝅]
[𝟎°, 𝟏𝟖𝟎°]
I and II
𝑦 = sec
𝑥
(−∞, −1] ∪ [1, ∞)
[𝟎°, 𝟗𝟎°) ∪ (𝟗𝟎°, 𝟏𝟖𝟎°]
I and II
−
𝟎,
𝝅
𝝅
∪
,𝝅
𝟐
𝟐