Trigonometric Functions and Their Properties

Graphing Tangent and Cotangent Functions

Understanding the graphs of tangent and cotangent functions is essential in Precalculus. These functions have unique properties, including vertical asymptotes and specific periods.

• Tangent Function: The general form is .

• Cotangent Function: The general form is .

Key Properties:

• Period of Tangent:

• Period of Cotangent:

• Vertical Asymptotes: Occur where the function is undefined.

Example 1: For :

• Period:

• Consecutive Asymptotes: and

• Key points: -intercept at , point at , point at

Example 2: For :

• Period:

• Consecutive Asymptotes: and

• Key points: -intercept at , point at , point at

Inverse Trigonometric Functions

Evaluating Inverse Trigonometric Expressions

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio. The principal values are typically restricted to specific intervals.

• arcsin or : Range:

• arccos or : Range:

• arctan or : Range:

Common Values Table:

Compositions and Algebraic Expressions with Inverse Trigonometric Functions

Compositions of Trigonometric and Inverse Trigonometric Functions

Compositions such as and can be rewritten as algebraic expressions in terms of .

• Example: , for

• Example: , for

Solving Trigonometric Equations

Quadratic Trigonometric Equations

Some trigonometric equations can be solved using algebraic techniques similar to those used for quadratic equations.

• Example:

• Let , then

• Solve for using the quadratic formula:

• Thus, or , so or

• Find values in the appropriate interval.

Trigonometric Identities and Angle Sum Formulas

Using Angle Sum and Difference Identities

Angle sum and difference identities allow us to find the sine, cosine, or tangent of sums or differences of angles.

• Sine Angle Sum:

• Cosine Angle Difference:

• Tangent Angle Sum:

Examples:

These identities are useful for evaluating trigonometric functions at non-standard angles and for simplifying expressions.