7.3 Double-Angle and Half-Angle Identities

Double-angle identities let you express trigonometric functions of 2θ in terms of θ. They're super handy for simplifying complex expressions and solving tricky equations. You'll use these a lot in trig, so get comfy with them!

Half-angle identities do the opposite, expressing trig functions of θ/2 in terms of θ. These are great for dealing with radicals and solving equations. Remember, the sign of your answer depends on which quadrant the angle's in.

Double-Angle Identities

Double-angle formulas for trigonometric functions

• $\sin(2\theta) = 2\sin\theta\cos\theta$ expresses double angle in terms of single angle sine and cosine
• Cosine double-angle offers three equivalent forms:
  • $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ utilizes difference of squares
  • $\cos(2\theta) = 2\cos^2\theta - 1$ eliminates sine term
  • $\cos(2\theta) = 1 - 2\sin^2\theta$ eliminates cosine term
• $\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$ relates double angle tangent to single angle tangent
• Derivation methods involve sum formulas and Pythagorean identity
• Applications include simplifying complex expressions (trigonometric ratios) and solving equations (finding angle values)

Half-Angle Identities

Half-angle formulas for trigonometric functions

• $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos\theta}{2}}$ expresses half-angle sine in terms of cosine
• $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1+\cos\theta}{2}}$ relates half-angle cosine to full angle cosine
• Tangent half-angle offers three equivalent forms:
  • $\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$ uses both positive and negative roots
  • $\tan(\frac{\theta}{2}) = \frac{\sin\theta}{1+\cos\theta}$ eliminates square root
  • $\tan(\frac{\theta}{2}) = \frac{1-\cos\theta}{\sin\theta}$ alternative form without square root
• Derivation methods use double-angle formulas and algebraic manipulation
• Sign selection depends on angle quadrant (positive in quadrants I and II, negative in III and IV)

Simplification with angle identities

• Strategies involve identifying identity opportunities and choosing most appropriate
• Common techniques include substituting double-angle formulas (reducing powers) and applying half-angle formulas (simplifying radicals)
• Verification requires checking domain restrictions (avoiding undefined values) and confirming equivalence (graphing or evaluating)

Equations using angle identities

• Strategies involve recognizing patterns matching formulas and substituting identities
• Solution methods:
  1. Algebraically manipulate equation
  2. Factor to isolate trigonometric term
  3. Solve resulting quadratic equations
• Verification includes checking solution ranges (0 to 2π or -π to π) and eliminating extraneous solutions
• Real-world applications found in physics (projectile motion) and engineering (signal processing)