Calculate sin(θ/2), cos(θ/2), and tan(θ/2) with automatic sign by quadrant, three tan forms, and double-angle back-verification.
The half-angle formulas express sin(θ/2), cos(θ/2), and tan(θ/2) using only the cosine and sine of the full angle θ. They are derived directly from the double-angle identities by substituting θ/2 for the variable and solving. The key formulas are: sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and tan(θ/2) has three equivalent forms: sin(θ/2)/cos(θ/2), sin θ/(1 + cos θ), and (1 − cos θ)/sin θ.
The ± sign in the sine and cosine formulas is determined by which quadrant θ/2 falls in. Since sin is positive in quadrants I and II and negative in III and IV, and cos is positive in I and IV but negative in II and III, you need to know where the half-angle lands to pick the correct sign.
This calculator automatically determines the quadrant of θ/2 and applies the correct sign. It computes all three forms of tan(θ/2), provides a visual quadrant map showing the current sign assignment, and verifies the results by reconstructing the original angle via the double-angle formulas. A comprehensive reference table of common angles from 0° to 360° rounds out the tool, making it invaluable for trigonometry courses, standardized test preparation, and engineering applications.
Half-Angle Formula Calculator helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like sin(θ/2), cos(θ/2), tan(θ/2) — Form 1 in one pass.
sin(θ/2) = ±√((1−cosθ)/2); cos(θ/2) = ±√((1+cosθ)/2); tan(θ/2) = sinθ/(1+cosθ) = (1−cosθ)/sinθ
Result: sin(30°) = 0.5, cos(30°) ≈ 0.8660, tan(30°) ≈ 0.5774
For θ = 60°: θ/2 = 30° (quadrant I, both signs positive). sin(30°) = √((1−cos60°)/2) = √((1−0.5)/2) = √0.25 = 0.5. cos(30°) = √((1+0.5)/2) = √0.75 ≈ 0.8660. tan(30°) = sin60°/(1+cos60°) = 0.8660/1.5 ≈ 0.5774.
This calculator is tailored to half-angle formula calculator workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
The sign depends on which quadrant θ/2 is in. Sin is positive in Q1 and Q2, negative in Q3 and Q4. Cos is positive in Q1 and Q4, negative in Q2 and Q3.
All three — sin(θ/2)/cos(θ/2), sin θ/(1+cos θ), and (1−cos θ)/sin θ — are algebraically equivalent but have different domains. The last two avoid square roots entirely.
Start with cos(2α) = 1 − 2sin²α, replace α with θ/2, and solve for sin(θ/2). Similarly for cosine, use cos(2α) = 2cos²α − 1.
Form 1 is undefined when cos(θ/2) = 0, i.e., θ/2 = 90° + n·180°. Form 2 fails when cos θ = −1 (θ = 180° + n·360°). Form 3 fails when sin θ = 0.
Setting t = tan(θ/2) transforms sin θ = 2t/(1+t²) and cos θ = (1−t²)/(1+t²), converting trigonometric integrals into rational function integrals. Use this as a practical reminder before finalizing the result.
They are inverses. The double-angle formulas express f(2θ) in terms of f(θ); half-angle formulas express f(θ/2) in terms of f(θ). You can derive one from the other.