Half Angle Calculator
Calculate half-angle formulas for sine, cosine, and tangent functions with exact values
Calculate Half-Angle Functions
Enter the full angle to calculate its half-angle functions
Half angle: 15.00° (0.2618 rad)
Quadrant: 1
Basic Functions for x = 30.0°
Half-Angle Functions for x/2 = 15.0°
Alternative Tangent Formulas (no ± sign)
Verification (Direct Calculation)
✓ All values match within computational precision
Example: Finding cos(15°)
Problem
Find the exact value of cos(15°) using half-angle formulas.
Solution
Since 15° = 30°/2, we can use the half-angle formula for cosine:
cos(15°) = cos(30°/2) = √[(1 + cos 30°)/2]
cos(30°) = √3/2
cos(15°) = √[(1 + √3/2)/2] = √[(2 + √3)/4]
cos(15°) = √(2 + √3)/2 ≈ 0.965926
Half-Angle Formulas
Sine Half-Angle
sin(x/2) = ±√[(1 - cos x)/2]
Cosine Half-Angle
cos(x/2) = ±√[(1 + cos x)/2]
Tangent Half-Angle
tan(x/2) = ±√[(1 - cos x)/(1 + cos x)]
tan(x/2) = (1 - cos x)/sin x
tan(x/2) = sin x/(1 + cos x)
Sign Determination
Sine (sin x/2)
+ in Q1, Q2 | - in Q3, Q4
Cosine (cos x/2)
+ in Q1, Q4 | - in Q2, Q3
Tangent (tan x/2)
+ in Q1, Q3 | - in Q2, Q4
Common Half-Angles
Understanding Half-Angle Formulas
What are Half-Angle Formulas?
Half-angle formulas allow you to find the exact values of trigonometric functions for an angle that is half of a known angle. They are derived from the power-reducing formulas and are particularly useful for finding exact values of angles like 15°, 22.5°, etc.
Why are they Important?
- •Find exact values for non-standard angles
- •Solve trigonometric equations
- •Simplify complex trigonometric expressions
- •Applications in calculus and physics
Derivation from Power-Reducing Formulas
Power-reducing formulas:
sin²(x) = (1 - cos(2x))/2
cos²(x) = (1 + cos(2x))/2
tan²(x) = (1 - cos(2x))/(1 + cos(2x))
By substituting x/2 for x and taking square roots, we get the half-angle formulas.
Key Point: The ± sign is determined by the quadrant of the half-angle, not the original angle.