Half Angle Calculator
Calculate trigonometric functions of half angles
How to Use This Calculator
Enter the Angle
Input the angle (θ) you want to calculate the half angle functions for.
Calculate
Click "Calculate Half Angle" to see sin(θ/2), cos(θ/2), and tan(θ/2) calculated both directly and using the formulas.
Note the Signs
Half angle formulas have ± signs. The calculator shows both positive and negative forms - use the quadrant to determine which sign is correct.
Formula
Half Angle Formulas:
sin(θ/2) = ±√[(1 - cos(θ))/2]
Use + in quadrants I & II, - in quadrants III & IV
cos(θ/2) = ±√[(1 + cos(θ))/2]
Use + in quadrants I & IV, - in quadrants II & III
tan(θ/2) = sin(θ)/(1 + cos(θ)) = (1 - cos(θ))/sin(θ)
Tangent half angle formula (no ± sign needed)
Sign Determination:
The ± sign depends on which quadrant θ/2 lies in. The calculator shows both forms - choose based on the quadrant of the half angle.
About Half Angle Calculator
The Half Angle Calculator helps you find trigonometric values for half angles (θ/2) using half angle identities. These formulas are derived from the double angle formulas and are essential for simplifying trigonometric expressions and solving equations.
When to Use This Calculator
- Trigonometric Simplification: Simplify expressions involving half angles
- Integration: Solve integrals using half angle substitution
- Problem Solving: Calculate half angle values from full angle values
- Verification: Verify manual calculations using half angle formulas
Why Use Our Calculator?
- ✅ Complete Results: Shows both positive and negative formula forms
- ✅ Formula Verification: Compares direct calculation with formula results
- ✅ Educational: Helps understand half angle identities
- ✅ 100% Free: No registration required
Frequently Asked Questions
How do I determine the sign in half angle formulas?
Determine which quadrant θ/2 lies in. For sin(θ/2): use + in quadrants I & II, - in III & IV. For cos(θ/2): use + in quadrants I & IV, - in II & III.
Why does tan(θ/2) not have a ± sign?
The tangent half angle formula tan(θ/2) = sin(θ)/(1 + cos(θ)) automatically gives the correct sign based on the values of sin(θ) and cos(θ), so no ± is needed.
Are half angle formulas the inverse of double angle formulas?
Yes! Half angle formulas are derived from double angle formulas by substitution. If you know cos(θ), you can find cos(θ/2) using the half angle formula.