Trigonometric Functions and Standard Values

Key Trigonometric Ratios

Trigonometric functions such as sine, cosine, and tangent are fundamental in relating the angles of a triangle to the lengths of its sides. Standard values for specific angles are frequently used in calculations and proofs.

• Sine Values: ,

• Cosine Values: ,

• Tangent Values: ,

Trigonometric Identities

Angle Addition and Subtraction Formulas

These identities allow the calculation of trigonometric functions for the sum or difference of two angles.

• Sine Addition:

• Cosine Addition:

• Tangent Addition:

Double Angle and Power-Reduction Formulas

These formulas are useful for simplifying expressions involving trigonometric functions of multiple angles.

• Cosine Double Angle:

• Sine Double Angle:

Trigonometric Function Transformations

Negative Angles and Co-function Identities

Trigonometric functions exhibit specific behaviors under angle transformations, such as negative angles or complementary angles.

• Negative Angle: ,

• Complementary Angle: ,

Graphs of Trigonometric Functions

Basic Graphs and Properties

The graphs of sine and cosine functions are periodic and exhibit symmetry. Understanding their shapes is essential for analyzing trigonometric equations and modeling periodic phenomena.

• Sine Function: Period , amplitude 1, symmetric about the origin (odd function).

• Cosine Function: Period , amplitude 1, symmetric about the y-axis (even function).

Summary Table: Key Trigonometric Identities

Applications and Examples

Example: Evaluating Trigonometric Expressions

Given and , find .

• Since

Additional info: The notes also contain some logarithmic and calculus content, but only trigonometric material is included here as per the course relevance.