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Unit Circle Trigonometry Overview

Sep 10, 2025

Overview

This lecture covers the unit circle, focusing on how to use it to evaluate sine and cosine functions for common angles in both degrees and radians.

The Unit Circle Basics

  • The unit circle is a circle with a radius of 1 centered at the origin.
  • Any point on the unit circle for angle θ has coordinates (cos θ, sin θ).
  • The x-value of a point corresponds to cos θ, and the y-value corresponds to sin θ.

Key Angles and Values in Quadrant I

  • At 30°: x = √3/2, y = 1/2.
  • At 45°: x = y = √2/2.
  • At 60°: x = 1/2, y = √3/2.
  • At 0°: (1, 0); at 90°: (0, 1).

Using the Unit Circle in Other Quadrants

  • For angles outside Quadrant I, use the reference angle and adjust signs based on quadrant:
    • Quadrant II: x negative, y positive.
    • Quadrant III: x and y negative.
    • Quadrant IV: x positive, y negative.

Example Evaluations (Degrees)

  • sin(45°) = √2/2 (use y-value at 45°).
  • sin(60°) = √3/2 (use y-value at 60°).
  • cos(180°) = –1 (use x-value at 180°).
  • sin(30°) = 1/2 (use y-value at 30°).
  • sin(135°) = √2/2 (reference angle 45°, x negative, y positive).
  • cos(225°) = –√2/2 (reference angle 45°, both x and y negative).
  • sin(315°) = –√2/2 (reference angle 45°, x positive, y negative).

Example Evaluations (Radians)

  • sin(Ï€/3) = √3/2 (use y-value at 60°).
  • cos(2Ï€/3) = –1/2 (reference angle 60°, x negative, y positive).
  • sin(4Ï€/3) = –√3/2 (reference angle 60°, both x and y negative).
  • cos(5Ï€/3) = 1/2 (reference angle 60°, x positive, y negative).
  • sin(Ï€/6) = 1/2 (use y-value at 30°).
  • cos(5Ï€/6) = –√3/2 (reference angle 30°, x negative, y positive).
  • cos(7Ï€/6) = –√3/2 (reference angle 30°, both x and y negative).
  • sin(11Ï€/6) = –1/2 (reference angle 30°, x positive, y negative).

Key Terms & Definitions

  • Unit Circle — A circle with radius 1, centered at (0,0) in the coordinate plane.
  • Reference Angle — The acute angle formed by the terminal side of an angle and the x-axis.
  • Sine (sin θ) — The y-coordinate of the point on the unit circle at angle θ.
  • Cosine (cos θ) — The x-coordinate of the point on the unit circle at angle θ.

Action Items / Next Steps

  • Memorize the x and y values for angles 30°, 45°, 60°, 0°, 90°, 180°, 270°, and 360°.
  • Practice evaluating sine and cosine for angles in all four quadrants using the unit circle.
  • Complete assigned homework or example problems involving unit circle trigonometry.

Full transcript