The half angle calculator above evaluates sin(θ/2), cos(θ/2) and tan(θ/2) from the half-angle identities, and — more usefully — tells you which way the ± in front of each square root resolves. The formulas themselves are short and easy to remember. The sign is where nearly everyone loses marks, because the radical only ever produces a positive number and the correct answer is frequently negative. This page shows the magnitude, the sign, the quadrant that decided it, and the reasoning, so the ± stops being a mystery.
Arb Digital publishes this alongside the rest of its trigonometry set because the half-angle identities are not just an exam exercise. They are the standard route to exact values for angles like 15°, 22.5° and 75° that do not appear on the unit circle, they turn awkward integrals of sin² and cos² into linear ones, and the tangent half-angle substitution converts an entire class of trigonometric integrals into rational functions. Getting the sign right matters everywhere those show up.
What This Half Angle Calculator Does
You enter an angle θ in degrees or radians. The tool halves it, works out which quadrant the half-angle falls in after reducing it to a single revolution, and then applies the three half-angle identities with the sign that quadrant demands. It reports all three functions, the half-angle itself, and the quadrant, and it flags the one genuine undefined case — a half-angle sitting exactly on 90° or 270°, where the tangent has a vertical asymptote.
That is a distinct job from the neighbouring tools on the site. Our double angle calculator runs the identity in the opposite direction, expressing sin 2θ and cos 2θ in terms of θ, and it has no sign ambiguity at all because nothing is square-rooted. Our trigonometric functions calculator simply evaluates all six functions at an angle you supply; if you only want the numerical value of sin 60°, use that one. This page exists specifically to demonstrate the identity route and the sign decision inside it.
How to Use It
- Enter θ, not θ/2. The input is the full angle. If you already have the half-angle and just want its sine, the trigonometric functions calculator is the faster tool.
- Pick the unit. Degrees by default. Selecting radians re-reads the number you typed as radians rather than converting it, so check the value after switching.
- Read the quadrant field first. It is the field that determines every sign on the page, and it is the one worth sanity-checking against your own reasoning.
- Compare against the exact surd. At six decimal places you can recognise 0.707107 as √2/2, 0.866025 as √3/2, and 0.258819 as the sine of 15°.
- Watch for the undefined tangent. When θ is an odd multiple of 180°, the half-angle sits on the vertical axis and tan(θ/2) does not exist.
The Formulas and How They're Calculated
All three identities descend from the double-angle formula for cosine, cos 2A = 1 − 2sin²A = 2cos²A − 1. Setting A = θ/2 and rearranging gives sin(θ/2) = ±√((1 − cos θ) ÷ 2) and cos(θ/2) = ±√((1 + cos θ) ÷ 2). Dividing the first by the second gives tan(θ/2) = ±√((1 − cos θ) ÷ (1 + cos θ)). The NIST Digital Library of Mathematical Functions section on trigonometric identities lists exactly these three with the ± printed, which is a good reminder that the ambiguity is part of the identity rather than an omission in a textbook.
Work the default value through. With θ = 120°, cos θ = −0.5, so the sine magnitude is √((1 − (−0.5)) ÷ 2) = √0.75 = 0.866025 and the cosine magnitude is √((1 + (−0.5)) ÷ 2) = √0.25 = 0.5. The half-angle is 60°, which is in quadrant I, so both signs are positive: sin 60° = 0.866025 and cos 60° = 0.5, which is correct. The tangent is 0.866025 ÷ 0.5 = 1.732051, or √3.
Now the case that catches people. Take θ = 480°. Its cosine is still −0.5, so the magnitudes are identical. But 480 ÷ 2 = 240°, which sits in quadrant III where sine and cosine are both negative. The correct answers are therefore sin 240° = −0.866025 and cos 240° = −0.5. Two angles with the same cosine gave half-angles with opposite signs. This is why you cannot read the sign off θ, and why reducing θ before halving it destroys the information you need.
The Sign Rule, Stated Properly
The rule is one sentence: determine which quadrant θ/2 lies in, then apply the ordinary sign convention for that quadrant. Sine is positive in quadrants I and II. Cosine is positive in quadrants I and IV. Tangent is positive in quadrants I and III. Nothing about the half-angle identity changes those conventions — the radical only ever hands you the magnitude, and the quadrant supplies the rest.
Two practical consequences follow. First, halving an angle roughly halves how far around the circle you are, so a θ in quadrant II lands its half in quadrant I, while a θ in quadrant IV lands its half in quadrant II. Second, and this is the trap, coterminal angles are not interchangeable here. 60° and 420° are the same angle on the circle, but their halves are 30° and 210°, which are in different quadrants with opposite signs. If a problem specifies a domain such as 0 ≤ θ < 720°, that upper bound exists precisely because the half-angle needs it. Our coterminal angle calculator and reference angle calculator handle the reduction step for the ordinary case where it is safe.
Why the Tangent Formula Escapes the Ambiguity
There is a genuinely useful piece of mathematics hiding here. Alongside the radical version, the tangent half-angle has two equivalent forms that carry no ± at all: tan(θ/2) = sin θ ÷ (1 + cos θ) and tan(θ/2) = (1 − cos θ) ÷ sin θ. Both are derived by multiplying the radical form by a conjugate, and the sign emerges automatically from the sign of sin θ.
Check it on the awkward case. With θ = 480°, sin θ = 0.866025 and 1 + cos θ = 0.5, so the first form gives 1.732051 — positive, which is right for quadrant III where tangent is positive. No quadrant reasoning was needed. The two forms have different failure points, which is why both are worth knowing: the first breaks when cos θ = −1 and the second breaks when sin θ = 0, so whenever one is undefined the other usually is not. This calculator uses these unambiguous forms internally and falls back to the radical only where they degenerate.
Getting Exact Values for 15°, 22.5° and 75°
The half-angle identities are the standard tool for extending the unit circle beyond the familiar 30-45-60 family. Halving 30° gives 15°: cos 30° = √3/2, so sin 15° = √((1 − √3/2) ÷ 2), which simplifies to (√6 − √2) ÷ 4, or 0.258819. Halving 45° gives 22.5°, where cos 22.5° = √(2 + √2) ÷ 2 = 0.923880. Halving 150° gives 75°, and halving 22.5° gives 11.25°, and the process never terminates — every angle produced can be halved again.
Type any of those into the tool and check the decimals against the surds. The nested radicals get ugly quickly, which is exactly why the numerical check is worth doing: an algebraic slip inside a nested square root is very hard to spot on the page but immediately obvious when the decimal disagrees at the third place. Paul's Online Notes at Lamar University covers the unit circle and how it is used to evaluate trigonometric functions, which is the reference frame these extended values slot into.
Where Half-Angle Formulas Actually Get Used
Three places, mostly. In integration, the rearranged forms sin²A = (1 − cos 2A) ÷ 2 and cos²A = (1 + cos 2A) ÷ 2 — the same identities read backwards — are what make ∫sin²x dx tractable, by turning a squared trigonometric function into a linear one. In calculus more broadly, the Weierstrass substitution t = tan(x/2) rewrites sin x as 2t/(1+t²) and cos x as (1−t²)/(1+t²), converting any rational function of sine and cosine into a rational function of t that partial fractions can finish.
Outside pure mathematics, half-angle relationships show up in geometry and physics whenever a chord, an inscribed angle or a reflection halves a central angle — the chord length of a circle is 2r·sin(θ/2), and Bragg's law in crystallography is written around the half-angle of the scattered beam for the same reason. In each of those settings the angle is physically constrained to a known range, which is why textbooks often present the formulas without the ± and why students are then surprised when a problem finally supplies an angle outside it. If you need the raw numeric evaluation at that point, the angle converter and the rest of our trigonometry set cover the arithmetic.
Arb Digital's content team turns complicated subject matter into pages that rank and actually teach, the way this one does.
Explore Content Marketing Talk To Our TeamCommon Mistakes to Avoid
- Reading the sign from θ instead of θ/2 — the quadrant that matters is the one the half-angle lands in, and it is very often a different one.
- Reducing θ to within 0–360° before halving — 420° and 60° are the same angle but their halves are 210° and 30°, with opposite signs for both sine and cosine.
- Assuming the radical output is the answer — a square root is never negative, so the formula alone can only ever give you a magnitude.
- Mixing degrees and radians — entering 1.5 while the unit selector says degrees gives a half-angle of 0.75°, not 0.75 radians, and the results differ enormously.
- Forgetting that tan(θ/2) can be undefined — when θ is an odd multiple of 180° the half-angle sits on the vertical axis and no finite tangent exists.
Related Free Tools From Arb Digital
Run the identity the other way with the double angle calculator, evaluate all six functions directly with the trigonometric functions calculator, strip an angle back to its acute equivalent with the reference angle calculator, find equivalent rotations with the coterminal angle calculator, or move between degrees, radians and gradians with the angle converter. Every mathematics tool we publish is listed on the free online tools hub.
Frequently Asked Questions
sin(θ/2) equals plus or minus the square root of (1 minus cos θ) over 2, cos(θ/2) equals plus or minus the square root of (1 plus cos θ) over 2, and tan(θ/2) equals plus or minus the square root of (1 minus cos θ) over (1 plus cos θ).
Work out which quadrant θ/2 falls in and apply the normal sign rules for that quadrant. Sine is positive in quadrants one and two, cosine in one and four, tangent in one and three.
Because halving moves the angle to a different position on the circle. An angle of 480 degrees and an angle of 120 degrees have the same cosine, but their halves land in quadrant three and quadrant one respectively, giving opposite signs.
Yes. tan(θ/2) equals sin θ divided by (1 plus cos θ), and it also equals (1 minus cos θ) divided by sin θ. Both produce the correct sign automatically, so no quadrant reasoning is needed.
Halve 30 degrees. With cos 30 equal to root three over two, the formula gives the square root of (1 minus root three over two) over 2, which simplifies to (root six minus root two) over four, about 0.258819.
When θ is an odd multiple of 180 degrees. The half-angle then sits exactly on 90 or 270 degrees, where cosine is zero and the tangent has a vertical asymptote.
Setting t equal to tan(x/2) rewrites sine and cosine as rational functions of t, which turns an integral of a rational expression in sine and cosine into an ordinary rational integral that partial fractions can handle.
This page explains a mathematical identity for educational purposes. Check any result that feeds into assessed or published work against your own derivation and the domain your problem specifies.
