Inverse Trigonometric Functions

Introduction to Inverse Functions

Inverse functions allow us to 'undo' the action of a function. In trigonometry, inverse trigonometric functions are used to find angles when given trigonometric values.

• Horizontal Line Test: A function has an inverse if and only if every horizontal line intersects its graph at most once (i.e., the function is one-to-one).

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

• Reflection: The graph of is a reflection of the graph of f about the line .

Inverse Sine Function

Definition and Properties

The inverse sine function, denoted as or , is the inverse of the sine function restricted to the domain .

• Domain:

• Range:

• Interpretation: means for in .

Graphing the Inverse Sine Function

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

• Alternatively, reflect the graph of the restricted sine function about the line .

Table: Common Values of

Example: Finding the Exact Value of an Inverse Sine

• Problem: Find the exact value of .

• Step 1: Let .

• Step 2: Rewrite as , where .

• Step 3: From the table, the angle in whose sine is is .

• Answer:

Applications

• Inverse trigonometric functions are used to solve for angles in right triangles when side lengths are known.

• They are also essential in solving equations involving trigonometric expressions and in modeling periodic phenomena.

Additional info: The notes continue with similar treatments for the inverse cosine and inverse tangent functions, including their definitions, domains, ranges, graphs, and example problems. These are standard topics in a Precalculus course.