Step-by-Step Solution

Half Angle Formulas

The half angle formulas are:

• \[ \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} \]
• \[ \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} \]
• \[ \tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)} \]

Properties:

• These formulas express trigonometric functions of half angles in terms of single angles
• The ± sign depends on the quadrant of θ/2
• Useful for simplifying expressions and solving equations
• Important in calculus and physics
• Can be derived from double angle formulas

Applications:

• Trigonometry
• Calculus (integration)
• Physics (wave motion)
• Engineering (signal processing)
• Computer graphics

Common Values:

• \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\)
• \(\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\)
• \(\tan(\frac{\pi}{4}) = 1\)
• \(\sin(\frac{\pi}{6}) = \frac{1}{2}\)
• \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
• \(\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\)