Half Angle Calculator
Calculate trigonometric functions of half angles using various formulas.
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:
- \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\)
- \(\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\)
- \(\tan(\frac{\pi}{4}) = 1\)
- \(\sin(\frac{\pi}{6}) = \frac{1}{2}\)
- \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
- \(\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\)