Special Angles

Overview

Special angles are commonly used in trigonometry and include 30°, 45°, and 60°. These angles are fundamental for solving problems involving the unit circle, trigonometric ratios, and right triangles. Recognizing their exact values in surd form is crucial for simplifying and solving problems.

---

Special Angles and Trigonometric Ratios

For $30^\circ$:

$$\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}$$

For $45^\circ$:

$$\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \tan 45^\circ = 1$$

For $60^\circ$:

$$\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \tan 60^\circ = \sqrt{3}$$

---

Use in the Unit Circle

• Definition: The unit circle is a circle with a radius of 1, centred at the origin of the coordinate plane.
• Special Angles on the Unit Circle:
  • The $x-coordinate$ represents $\cos \theta$
  • The $y-coordinat$e represents $\sin \theta$

---

Worked Examples

---

---

Summary

• Special Angles: 30°, 45°, and 60° are key in trigonometry.
• Trigonometric Ratios in Surd Form: Memorize $\sin, \cos$, and ⁡$\tan$ values for these angles.
• Unit Circle: Provides a framework for understanding angles and their trigonometric values. These angles and their properties simplify calculations and are essential for problem-solving in trigonometry.