Lecture Notes: The Unit Circle and Trigonometric Ratios

Introduction

Focus on the unit circle and key trigonometric ratios for IB exams (non-calculator).
Importance of memorizing two 'magic triangles'.

Purpose: Help to find sine, cosine, and tangent of key angles.
Triangles Explained:
1. 45° Triangle:
  - 45° in radians is π/4.
  - Sine of 45°: Opposite/Hypotenuse = 1/√2.
  - Memorize this for non-calculator environments.
2. 30° and 60° Triangle:
  - 30° in radians is π/6.
  - Cosine of 30°: Adjacent/Hypotenuse = √3/2.
  - Tangent of 60°: Opposite/Adjacent = √3.
Using these triangles helps to calculate trigonometric ratios for angles 30°, 45°, and 60°.

Definition: A circle with a radius of 1 centered at the origin (0,0).
Key Points on Unit Circle:
- Coordinates of points are based on a radius of 1.
- Important angle measures: 0° (0 radians), 90° (π/2), 180° (π), 270° (3π/2), and 360° (2π).
Example: Sine of 30°
- Sine of 30° = 1/2 = 0.5 (height of the line from the circle).
Cosine of 30°
- Cosine of 30° = √3/2 = ~0.85 (width of line from the circle).
- Shows relation of angle components to the unit circle.

Quadrants and Signs:
- First Quadrant (0° to 90°):
  - All trigonometric functions (sine, cosine, tangent) are positive.
- Second Quadrant (90° to 180°):
  - Only sine is positive.
- Third Quadrant (180° to 270°):
  - Only tangent is positive.
- Fourth Quadrant (270° to 360°):
  - Only cosine is positive.
Acronym: ASTC – All, Sine, Tangent, Cosine (for remembering the positives in each quadrant).

Learn to calculate sine, cosine, and tangent of 30°, 45°, and 60° in any quadrant using these methods.
Encouragement to practice and understand the unit circle to become comfortable with these calculations.