Half Angle Calculator

Our Half Angle Calculator helps you find the trigonometric functions (sine, cosine, and tangent) of half an angle using half-angle formulas. This is particularly useful in:

• Trigonometry calculations
• Mathematical analysis
• Engineering problems
• Physics equations
• Geometric constructions

How to Use the Calculator

1. Enter your angle in degrees
2. Get instant results for:
   • Original angle’s trigonometric functions (sin, cos, tan)
   • Half angle’s trigonometric functions (sin(x/2), cos(x/2), tan(x/2))

Understanding Half-Angle Formulas

The calculator uses these standard half-angle formulas:

• sin(x/2) = ±√[(1 - cos(x))/2]
• cos(x/2) = ±√[(1 + cos(x))/2]
• tan(x/2) = ±√[(1 - cos(x))/(1 + cos(x))]

The sign (+ or -) depends on which quadrant the half angle falls in:

• Sine: Positive in quadrants 1 and 2, negative in quadrants 3 and 4
• Cosine: Positive in quadrants 1 and 4, negative in quadrants 2 and 3
• Tangent: Positive in quadrants 1 and 3, negative in quadrants 2 and 4

Common Applications

Mathematics

• Solving trigonometric equations
• Verifying trigonometric identities
• Geometric proofs
• Complex number calculations

Physics & Engineering

• Wave analysis
• Angular motion calculations
• Signal processing
• Mechanical systems

Computer Graphics

• Rotation calculations
• Animation
• Game development
• 3D modeling

Tips for Accurate Results

1. Input Angle

   • Use degrees for angle measurement
   • Make sure the angle is between -360° and 360°
   • For best results, use decimal points if needed

2. Understanding Results

   • Results are rounded to 4 decimal places
   • Pay attention to the signs based on quadrants
   • Remember that half angles have specific domains

Technical Notes

• The calculator uses high-precision arithmetic for accurate results
• All calculations are performed using standard mathematical libraries
• Results are validated against known trigonometric values
• Special cases (like 30°, 45°, 60°) give exact values where applicable

Frequently Asked Questions

What is cos(15°)?

cos(15°) = √[(2 + √3)/4] ≈ 0.9659. This is because 15° is half of 30°, and we can use the half-angle formula with cos(30°) = √3/2.

Is cos(x/2) the same as cos(x)/2?

No, they are different. cos(x/2) is the cosine of half the angle, while cos(x)/2 is half of the cosine of the angle. For example, cos(60°)/2 = 0.25, but cos(30°) = 0.866.

Why do we need half-angle formulas?

Half-angle formulas are essential in many mathematical and practical applications, from solving complex trigonometric equations to engineering calculations involving rotational motion.