Inverse Trigonometric Functions

Introduction

Inverse trigonometric functions are essential tools in precalculus, allowing us to determine angles when given trigonometric values. These functions are defined as the inverses of the basic trigonometric functions, but with restricted domains to ensure they are one-to-one and thus invertible.

Inverse Functions

Definition and Properties

• One-to-One Functions: A function has an inverse if and only if it is one-to-one, meaning no horizontal line intersects its graph more than once.

• Graphical Relationship: If the point (a, b) is on the graph of f, then (b, a) is on the graph of its inverse f-1.

• Reflection: The graph of an inverse function is a reflection of the original function about the line .

The Inverse Sine Function

Definition and Domain

• The sine function is not one-to-one over its entire domain, so it is restricted to to define its inverse.

• The inverse sine function, denoted or , satisfies for and .

Graphing the Inverse Sine Function

• To graph , reverse the coordinates of points on (restricted domain), or reflect the graph about .

Finding Exact Values

• Let , then .

• Use known values from the unit circle to find exact values.

Example: Find . Let . Then , so .

The Inverse Cosine Function

Definition and Domain

• The cosine function is not one-to-one over its entire domain, so it is restricted to to define its inverse.

• The inverse cosine function, denoted or , satisfies for and .

Graphing the Inverse Cosine Function

• To graph , reverse the coordinates of points on (restricted domain), or reflect the graph about .

Finding Exact Values

• Let , then .

• Use known values from the unit circle to find exact values.

Example: Find . Let . Then , so .

The Inverse Tangent Function

Definition and Domain

• The tangent function is not one-to-one over its entire domain, so it is restricted to to define its inverse.

• The inverse tangent function, denoted or , satisfies for and .

Graphing the Inverse Tangent Function

• To graph , reverse the coordinates of points on (restricted domain), or reflect the graph about .

Finding Exact Values

• Let , then .

• Use known values from the unit circle to find exact values.

Example: Find . Let . Then , so .

The Inverse Cotangent, Cosecant, and Secant Functions

Definitions

• Inverse Cotangent: is the inverse of the restricted cotangent function.

• Inverse Cosecant: is the inverse of the restricted cosecant function.

• Inverse Secant: is the inverse of the restricted secant function.

These functions are less commonly used but are defined similarly by restricting the domains of their respective trigonometric functions to ensure they are one-to-one.

Using Calculators with Inverse Trigonometric Functions

Evaluating Values

• Most scientific calculators have keys for , , and .

• To find values to four decimal places, enter the value and use the appropriate inverse function key.

Example: Use a calculator to find . The result is approximately radians.

Inverse Properties

Relationships Between Functions and Their Inverses

• Sine and Inverse Sine: for and for .

• Cosine and Inverse Cosine: for and for .

• Tangent and Inverse Tangent: for all real and for .

Evaluating Compositions of Functions and Their Inverses

Examples

• Find the exact value of . Since returns an angle whose sine is , the answer is $0.6$.

• Find the exact value of . Since $2[0, \pi].

• If the value is not in the domain of the inverse function, the expression is not defined.