Power-Reducing and Half-Angle Formulas

Sine Squared Half-Angle Formula

• Power-Reducing Formula: $ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} $
• Half-Angle Formula: $ \sin^2(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{2} $
  • Taking the square root of both sides: $ \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} $

Cosine Squared Half-Angle Formula

• Power-Reducing Formula: $ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} $
• Half-Angle Formula: $ \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} $

Tangent Half-Angle Formulas

• Primary Formula: $ \tan(\frac{\theta}{2}) = \frac{\sin(\theta/2)}{\cos(\theta/2)} $
• Using Sine and Cosine Half-Angle Formulas:
  • $ \tan(\frac{\theta}{2}) = \frac{\sqrt{1 - \cos(\theta)/2}}{\sqrt{1 + \cos(\theta)/2}} $
  • Simplifies to: $ \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} $
  • Other forms:
    • $ = \frac{1 - \cos(\theta)}{\sin(\theta)} $
    • $ = \frac{\sin(\theta)}{1 + \cos(\theta)} $

Example Problems

Example 1: Evaluate Cosine of 15 Degrees

• Formula: $ \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} $
• Given $ \theta/2 = 15^ ext{°} $, so $ \theta = 30^ ext{°} $
  • $ \cos(15^ ext{°}) = \pm\sqrt{\frac{1 + \cos(30^ ext{°})}{2}} $
  • Known: $ \cos(30^ ext{°}) = \frac{\sqrt{3}}{2} $
  • Simplification:
    • $ \cos(15^ ext{°}) = \sqrt{2 + \sqrt{3}/2} $
  • Result:
    • $ = \frac{\sqrt{2 + \sqrt{3}}}{2} $

Example 2: Evaluate Sine of 22.5 Degrees

• Formula: $ \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} $
• Given $ \theta/2 = 22.5^ ext{°} $, so $ \theta = 45^ ext{°} $
  • $ \sin(22.5^ ext{°}) = \pm\sqrt{\frac{1 - \cos(45^ ext{°})}{2}} $
  • Known:
    • $ \cos(45^ ext{°}) = \frac{\sqrt{2}}{2} $
  • Simplification:
    • $ \sin(22.5^ ext{°}) = \sqrt{2 - \sqrt{2}/2} $
  • Result:
    • $ = \frac{\sqrt{2 - \sqrt{2}}}{2} $
  • Decimal approximation:
    • $ \sin(22.5^ ext{°}) \approx 0.3827 $

Example 3: Evaluate Tangent of 75 Degrees

• Formula: $ \tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)} $
• Given $ \theta/2 = 75^ ext{°} $, so $ \theta = 150^ ext{°} $
  • $ \tan(75^ ext{°}) = \frac{1 - \cos(150^ ext{°})}{\sin(150^ ext{°})} $
  • Known:
    • $ \cos(150^ ext{°}) = -\frac{\sqrt{3}}{2} $
    • $ \sin(150^ ext{°}) = \frac{1}{2} $
  • Simplification:
    • $ \tan(75^ ext{°}) = \frac{1 + \sqrt{3}/2}{1/2} \times 2 $
  • Result:
    • $ = 2 + \sqrt{3} $