Working through a trig identity and need to find sine, cosine, or tangent of half an angle? This half angle formula calculator computes all six trigonometric functions for θ/2 given any input angle θ.
Half Angle Calculator
Using the half angle calculator
Enter your angle in the Angle (θ) field and choose Degrees or Radians from the dropdown. The calculator instantly shows all six trigonometric functions for half that angle: sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
The result title shows you the computed angle – enter 60° and you'll see results for 30°.
What are the half angle formulas?
The half angle formulas express trigonometric functions of θ/2 in terms of trigonometric functions of θ. They're derived from the double angle formulas by substitution.
Half angle formula for sine
Half angle formula for cosine
Half angle formula for tangent
Tangent has multiple equivalent forms (lucky you!):
The ± sign in the sine and cosine formulas depends on which quadrant θ/2 falls in. For angles where θ/2 is in the first quadrant (0° to 90°), both sine and cosine are positive.
Example: half angle of 60°
Let's compute sin(30°) using the half angle formula with θ = 60°:
This matches the known value of sin(30°) = 1/2.
Deriving the half angle formulas
The half angle formulas come from the cosine double angle formula. Starting with:
Solve for sin(α):
Now substitute α = θ/2, so 2α = θ:
Take the square root of both sides to get the half angle sine formula. The cosine formula follows similarly from cos(2α) = 2cos²(α) - 1.
When to use half angle formulas
Half angle formulas are useful when you:
- Know a trig value for θ but need the value for θ/2
- Need to find exact values for angles like
15°or22.5°(half of30°or45°) - Are simplifying integrals involving even powers of sine or cosine
- Need to convert between different trigonometric expressions
