Trigonometric Functions and Standard Values

Key Trigonometric Ratios

Trigonometric functions relate the angles of a triangle to the ratios of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). For certain standard angles, these functions have well-known values that are frequently used in calculations.

• sin(37°) = \frac{3}{5}

• sin(53°) = \frac{4}{5}

• cos(37°) = \frac{4}{5}

• cos(53°) = \frac{3}{5}

• tan(37°) = \frac{3}{4}

• tan(53°) = \frac{4}{3}

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-Reducing Formulas

These formulas are useful for simplifying expressions involving trigonometric functions of double angles or powers of functions.

• Cosine Double Angle:

Trigonometric Function Transformations

Sign Change and Reference Angle Properties

Trigonometric functions exhibit specific behaviors under angle transformations, such as sign changes and reference angles.

• sin(180° - θ) = sin θ

• cos(180° - θ) = -cos θ

Graphs of Trigonometric Functions

Basic Graphs and Properties

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

• Sine and Cosine Graphs: Both functions have a period of and amplitude of 1.

• Key Points: , , ; , , .

Summary Table: Standard Trigonometric Values

Additional info:

• Some images and equations in the source also reference logarithms and calculus concepts, but only trigonometric content is included here as per the course relevance.