Inverse Trigonometric Functions
Inverse Trigonometric Functions
Concept: Definition
Inverse trigonometric functions are functions that undo the trigonometric functions.
Concept: Notation
The inverse trigonometric functions are denoted by:
sin^(1)(x)
cos^(1)(x)
tan^(1)(x)
cot^(1)(x)
sec^(1)(x)
csc^(1)(x)
Concept: Domain and Range
Function  Domain  Range 

sin^(1)(x) 
[1, 1]  [π/2, π/2] 
cos^(1)(x) 
[1, 1]  [0, π] 
tan^(1)(x) 
All real numbers  (π/2, π/2) 
cot^(1)(x) 
All real numbers except 0  (0, π) 
sec^(1)(x) 
(∞, 1] ∪ [1, ∞)  [0, π/2) ∪ (π/2, π] 
csc^(1)(x) 
(∞, 1] ∪ [1, ∞)  [π/2, 0) ∪ (0, π/2] 
Concept: Graphs

sin^(1)(x):
 Starts at (1, π/2)
 Ends at (1, π/2)

cos^(1)(x):
 Starts at (1, π)
 Ends at (1, 0)

tan^(1)(x):
 Starts at (∞, π/2)
 Ends at (∞, π/2)

cot^(1)(x):
 Starts at (∞, 0)
 Ends at (∞, π)

sec^(1)(x):
 Starts at (∞, π/2)
 Ends at (1, π)
 Starts again at (1, 0)
 Ends at (∞, π/2)

csc^(1)(x):
 Starts at (∞, π/2)
 Ends at (1, 0)
 Starts again at (1, π/2)
 Ends at (∞, π/2)
Concept: Identities
sin^(1)(x) = cos^(1)√(1  x^2)
cos^(1)(x) = sin^(1)√(1  x^2)
tan^(1)(x) = cot^(1)(1/x)
cot^(1)(x) = tan^(1)(1/x)
sec^(1)(x) = cos^(1)(1/x)
csc^(1)(x) = sin^(1)(1/x)
Concept: Applications
Inverse trigonometric functions are used in a variety of applications, including:
 Solving trigonometric equations
 Finding the angle between two lines
 Finding the area of a triangle
 Finding the direction of a vector