Inverse Trigonometric Functions

Introduction to Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. These functions are essential in solving equations involving trigonometric expressions and in applications such as geometry, physics, and engineering.

• Inverse Sine Function (arcsin or )

• Inverse Cosine Function (arccos or )

• Inverse Tangent Function (arctan or )

Key objectives include understanding and using these functions, evaluating them with calculators, and finding exact values of composite functions involving inverse trigonometric functions.

Review of Inverse Functions

• Horizontal Line Test: A function is one-to-one (and thus has an inverse) if no horizontal line intersects its graph more than once.

• Graphical Relationship: If the point is on the graph of , then is on the graph of . The graph of is a reflection of the graph of about the line .

Inverse Sine Function

Definition and Properties

The sine function is not one-to-one over its entire domain, so its inverse is defined by restricting the domain to .

• Inverse Sine Function: Denoted or .

• Domain:

• Range:

• Interpretation: means for in .

Graphing the Inverse Sine Function

• To graph , take points from the graph of (with restricted domain) and reverse the order of the coordinates.

• Alternatively, reflect the graph of (restricted) about the line .

Table: Key Values of

Example: Finding the Exact Value of an Inverse Sine Function

Find the exact value of .

1. Let .

2. Rewrite as , where .

3. From the table, the angle in whose sine is is .

Answer:

Additional info: The same process applies for other values in the table, using the restricted range for the inverse sine function.