Inverse Trigonometric Functions
Understanding Inverse Trigonometric Functions
For a function to have an inverse, it must be one-to-one, meaning each output is mapped from a unique input. However, trigonometric functions are periodic and not inherently one-to-one. To make them invertible, their domains are restricted to specific intervals where they are one-to-one. These restricted functions allow us to define their inverses, ensuring that each output corresponds to a unique input.
Inverse trigonometric functions, also known as arcus functions or cyclometric functions, are the inverses of the six primary trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. They are used to determine angles based on known trigonometric values and have applications in fields like geometry, engineering, and physics.
Notation
- Inverse sine:
or - Inverse cosine:
or - Inverse tangent:
or - Inverse cotangent:
or - Inverse secant:
or - Inverse cosecant:
or
Key Properties and Formulas
Negation Identities
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Reciprocal Relationships
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Complementary Identities
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,
Sum and Difference Formulas
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if -
if
Double and Triple Angle Formulas
Graphs of Inverse Trigonometric Functions
Each inverse trigonometric function has a distinct graph that helps visualize its behavior:
- Arcsine Function (
): Defined for , with range .
- Arccosine Function (
): Defined for , with range .
- Arctangent Function (
): Defined for all , with range .
- Arccotangent Function (
): Defined for all , with range .
- Arcsecant Function (
): Defined for , with range .
- Arccosecant Function (
): Defined for , with range .
Summary
| Inverse Trigonometric Function | Domain | Range |
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for . -
for . -
is different from , which represents . - Inverse trigonometric functions are essential in solving trigonometric equations and finding angles in applied problems.