Half-Angle Calculator — sin, cos and tan of θ/2
Enter any angle θ in degrees or radians and instantly get sin(θ/2), cos(θ/2) and tan(θ/2) using the half-angle formulas derived from the double-angle identities for cosine.
Angle unit
Sign of θ/2
Half-angle sine: ±√((1 − cos θ) / 2)
- 1
Convert θ to radians
60° × π ÷ 180 = 1.047198 - 2
cos θ
cos(1.047198 rad) = 0.5 - 3
sin²(θ/2) = (1 − cos θ) ÷ 2
(1 − 0.5) ÷ 2 = 0.25 - 4
sin(θ/2) = ±√(sin²(θ/2))
+√0.25 = 0.500000
How does this calculator work?
Half-angle formulas give trig ratios at θ/2 from the parent angle θ: sin(θ/2) = ±√((1−cosθ)/2), cos(θ/2) = ±√((1+cosθ)/2), tan(θ/2) = sinθ/(1+cosθ). The ± depends on the quadrant of θ/2. Enter θ in degrees or radians and select the sign to get all three values instantly.
Formula
How this is calculated
The half-angle formulas are derived from the double-angle identities for cosine. Because cos(2α) = 1 − 2sin²α, substituting α = θ/2 gives sin²(θ/2) = (1 − cos θ)/2, so sin(θ/2) = ±√((1 − cos θ)/2). Similarly, cos(2α) = 2cos²α − 1 rearranges to cos(θ/2) = ±√((1 + cos θ)/2). The tangent follows from the ratio, but the algebraically equivalent form tan(θ/2) = sin θ / (1 + cos θ) avoids indeterminate forms when the denominator of sin/cos is zero.
The ± sign on sin and cos depends on the quadrant of θ/2, not of θ itself. If θ = 300°, then θ/2 = 150° which is in the second quadrant — where sine is positive but cosine is negative. The sign selector lets you specify this correctly. For tan(θ/2) the sign is determined automatically since it derives directly from sin θ and cos θ.
The calculator evaluates formulas numerically using 64-bit floating-point arithmetic, so tiny rounding residuals may appear near special angles (e.g., cos(180°/2) ≈ 6.1e-17 rather than exactly 0). For exact symbolic results use a computer algebra system. The computation is undefined when 1 + cos θ ≈ 0 (near θ = ±180°) because tan(θ/2) diverges there.
Frequently asked questions
Half-angle formulas express trig ratios of θ/2 in terms of the full angle θ: sin(θ/2) = ±√((1−cosθ)/2), cos(θ/2) = ±√((1+cosθ)/2), tan(θ/2) = sinθ/(1+cosθ). They come directly from the double-angle identity cos(2α) = 1−2sin²α.
Determine which quadrant θ/2 falls in. Quadrant I (0°–90°): both sin and cos positive. Quadrant II (90°–180°): sin positive, cos negative. Quadrant III (180°–270°): both negative. Quadrant IV (270°–360°): sin negative, cos positive. Set the sign selector accordingly.
Both expressions are equivalent, but sin θ / (1 + cos θ) avoids introducing a separate ± ambiguity and stays numerically stable except near θ = ±180° where tan(θ/2) itself is undefined. The formula using sin/cos of the half angle would also require you to pick consistent signs.
Also known as
TG we-Calculate Editorial Team. (2026). Half-Angle Calculator — sin, cos and tan of θ/2 [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/half-angle-calculator
TG we-Calculate Editorial Team. "Half-Angle Calculator — sin, cos and tan of θ/2." TG we-Calculate. 2026. https://we-calculate.com/calculator/half-angle-calculator.
TG we-Calculate Editorial Team, "Half-Angle Calculator — sin, cos and tan of θ/2," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/half-angle-calculator
@misc{wecalculate_half_angle_calculator, title = {Half-Angle Calculator — sin, cos and tan of θ/2}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/half-angle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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