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The half-angle identities express sin(θ/2), cos(θ/2), and tan(θ/2) purely in terms of cos(θ) — meaning you can find the sine or cosine of half an angle without ever knowing θ/2's own reference angle. They technically involve a ± sign in their textbook form, since θ/2 could in principle fall in either of two half-planes — but because you give this calculator an exact θ, θ/2 is fully and unambiguously determined, and the correct sign is resolved automatically rather than left for you to figure out. These identities are derived directly from the power-reducing (cosine double-angle) identities by substituting θ/2 in place of θ, and they're commonly used in calculus (certain trig substitutions and integrals simplify only after a half-angle rewrite) and in engineering fields that work with half-wave rectified signals or bisected angles in geometry and design.
How it works
Enter the angle θ. The calculator first finds cos(θ), then applies the half-angle formulas: sin(θ/2) = ±√((1−cosθ)/2) and cos(θ/2) = ±√((1+cosθ)/2), with tan(θ/2) available either as their ratio or via the equivalent identity tan(θ/2) = sinθ/(1+cosθ). To resolve the ambiguous ± sign, the calculator computes θ/2 directly from your input θ and checks which quadrant that half-angle actually falls in, then applies whichever sign (positive or negative) matches sine and cosine's known signs in that quadrant.
- Enter angle (θ).
- Enter angle unit.
- Click Calculate to see your results.
Examples
θ = 60°
sin(60°/2) = sin(30°) = 0.5, matching the half-angle formula √((1−cos60°)/2) = √((1−0.5)/2) = √0.25 = 0.5.
θ = 90°
cos(90°/2) = cos(45°) = √2/2 ≈ 0.7071, matching √((1+cos90°)/2) = √((1+0)/2) = √0.5 ≈ 0.7071.
θ = 120°
sin(120°/2) = sin(60°) ≈ 0.8660, matching √((1−cos120°)/2) = √((1−(−0.5))/2) = √0.75 ≈ 0.8660.
Who should use it
- Calculus substitutions where a half-angle rewrite simplifies an integral.
- Trigonometry coursework on the half-angle identities.
- Geometry problems involving a bisected angle.
Industry applications
- Calculus and trigonometry education
- Signal processing (half-wave rectification analysis)
Advantages
- Resolves the ambiguous ± sign automatically instead of leaving it to you.
- Works for any angle θ, not just the standard textbook reference angles.
Limitations
- tan(θ/2) has an undefined edge case when cosθ = −1, which needs special handling.
Common mistakes to avoid
- Assuming the + sign always applies — it actually depends on which quadrant θ/2 falls in, not which quadrant θ itself is in.
- Forgetting the square root in the sin(θ/2) and cos(θ/2) formulas and using (1−cosθ)/2 directly as if it were the final answer.
- Mixing up which formula has the + and which has the − sign inside the square root (cosine uses +cosθ, sine uses −cosθ).
Best practices
- Compute θ/2 first and identify its quadrant before finalizing the sign, rather than guessing based on θ's own quadrant.
- Use the sinθ/(1+cosθ) form for tangent when you want to avoid an extra sign decision.
- Cross-check your half-angle result against a direct evaluation (e.g., using the Sine Calculator on θ/2 itself) when the half-angle happens to be a common reference angle.
Tips
- Need the reverse operation — going from θ to 2θ — instead? Use the Double Angle Formula Calculator.