Pythagorean Identities

The Pythagorean identities are fundamental relationships in trigonometry, derived from the geometry of the unit circle. This section focuses on these identities, double angle identities, and their applications.

Key Concepts

• Pythagorean Identity:
  • \(\cos^2\theta + \sin^2\theta = 1\).
• Double Angle Identities:
  • \(\sin(2\theta) = 2\sin\theta\cos\theta\).
  • \(\cos(2\theta) = \cos^2\theta – \sin^2\theta = 2\cos^2\theta – 1 = 1 – 2\sin^2\theta\).
• Relationships Between Trigonometric Ratios:
  • Use algebraic manipulation and Pythagorean identities to find relationships between \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) without explicitly calculating \(\theta\).

Guidance, Clarifications, and Syllabus Links

• Illustrations: Use geometrical diagrams and dynamic graphing software to visualize the relationships between trigonometric functions and identities.
• Applications:
  • Verify identities using algebraic techniques.
  • Derive relationships between trigonometric functions for specific cases.
• Focus: Solve problems using identities, especially for simplifying expressions and verifying equations.

Examples

Example 1: Using the Pythagorean Identity

• Given \(\sin\theta = \frac{3}{5}\), find \(\cos\theta\):
  • \(\cos^2\theta = 1 – \sin^2\theta = 1 – \left(\frac{3}{5}\right)^2 = 1 – \frac{9}{25} = \frac{16}{25}\).
  • \(\cos\theta = \pm\frac{4}{5}\) (choose the sign based on the quadrant).

Example 2: Double Angle Identity

• Given \(\cos x = \frac{3}{4}\) and \(x\) is acute, find \(\sin(2x)\):
  • \(\sin x = \sqrt{1 – \cos^2x} = \sqrt{1 – \left(\frac{3}{4}\right)^2} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}\).
  • \(\sin(2x) = 2\sin x \cos x = 2 \cdot \frac{\sqrt{7}}{4} \cdot \frac{3}{4} = \frac{3\sqrt{7}}{8}\).