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
