Trigonometric Functions and Identities

Basic Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions are sine, cosine, and tangent, commonly abbreviated as sin, cos, and tan.

• Sine (sin θ): Ratio of the length of the side opposite angle θ to the hypotenuse.

• Cosine (cos θ): Ratio of the length of the adjacent side to the hypotenuse.

• Tangent (tan θ): Ratio of the length of the opposite side to the adjacent side.

Example: If and is in the first quadrant, then and .

Pythagorean Identity

The Pythagorean identity is a fundamental relationship among the trigonometric functions:

This identity is used to find one trigonometric value given another.

Reciprocal and Quotient Identities

Solving Trigonometric Equations

Solving for Angles

To solve equations such as or , use the inverse trigonometric functions and consider the periodicity of the functions.

• For , solutions are and , where is any integer.

• For , solutions are and .

Example: has solutions and .

Quadratic Trigonometric Equations

Equations such as can be solved by setting each factor to zero and solving for .

• For ,

• For , or

Trigonometric Identities and Exact Values

Sum and Difference Formulas

These formulas allow the calculation of trigonometric functions of sums or differences of angles:

Example:

Double Angle and Half Angle Formulas

Example: If and is in quadrant IV, then and can be found using the double angle formulas.

Inverse Trigonometric Functions

Definitions and Properties

Inverse trigonometric functions return the angle whose trigonometric function equals a given value.

• means and

• means and

• means and

Example:

Solving Trigonometric Equations in Intervals

Finding All Solutions

To find all solutions in a given interval, use the periodicity of the trigonometric functions and the principal values from the inverse functions.

• For in , use and .

Applications and Problem Solving

Using Sketches and Reference Triangles

Reference triangles and sketches help determine the sign and value of trigonometric functions in different quadrants.

• Given in quadrant IV, is negative and is negative.

Evaluating Trigonometric Expressions

Use identities and known values to evaluate expressions such as or .

Summary Table: Key Trigonometric Identities

Additional info:

• Some problems require using the unit circle and knowledge of reference angles.

• Solving trigonometric equations often involves considering all possible solutions within a specified interval due to the periodic nature of the functions.

• Inverse trigonometric functions are used to find exact values of angles given a trigonometric ratio.