Trigonometric Periodicity and Coterminal Angles

Overview

This lecture covers the periodic properties of trigonometric functions, how coterminal angles affect their values, and provides worked examples of evaluating trigonometric expressions using these properties.

Periodic Properties of Trigonometric Functions

• Trigonometric functions repeat their values in regular intervals called periods.
• For any angle θ, sine(θ) = sine(θ + 360°) = sine(θ + 2π).
• Cosine and all other trig functions (tangent, cotangent) are also periodic with period 360° or 2π.
• Coterminal angles differ by multiples of 360° (or 2π radians) and have equal trigonometric values.
• Graphs of sine and cosine show this repeating pattern or periodicity.

Examples Using Periodic Properties

• sine(30°) = sine(390°) because 390 - 30 = 360°, so they are coterminal.
• To evaluate sine(420°), subtract 360°: 420 - 360 = 60°, thus sine(420°) = sine(60°) = √3/2.
• To evaluate cosine(-2π/3), add 2π: -2π/3 + 2π = 4π/3, so cosine(-2π/3) = cosine(4π/3).
• At 4π/3, both x and y coordinates are negative on the unit circle, so cosine(4π/3) = -1/2.
• To evaluate sine(750°), subtract 360° twice: 750 - 360 = 390, then 390 - 360 = 30, so sine(750°) = sine(30°) = 1/2.

Key Terms & Definitions

• Periodic Function — A function whose values repeat at regular intervals.
• Coterminal Angles — Angles that differ by multiples of 360° (or 2π radians) and share the same trigonometric values.
• Reference Angle — The acute angle formed by the terminal side of an angle and the x-axis.
• Unit Circle — A circle with radius 1 used to define trigonometric values at various angles.

Action Items / Next Steps

• Practice finding coterminal angles and evaluating trig functions for large angle measures.
• Review the unit circle and memorize key sine and cosine values.