Half Angle Calculator
Enter any angle theta and this calculator instantly applies the half-angle identities to find all six trigonometric values at theta/2. Choose degrees or radians, and the sign of each result is resolved automatically from the quadrant of the half angle. The "Show your work" panel walks through every substitution so you can follow along or check your homework.
Formula
Worked example
For theta = 30 deg: cos(30 deg) = sqrt(3)/2 ~ 0.866025. sin(15 deg) = sqrt[(1 - 0.866025)/2] = sqrt(0.066987) ~ 0.258819. cos(15 deg) = sqrt[(1 + 0.866025)/2] = sqrt(0.933013) ~ 0.965926. tan(15 deg) = (1 - 0.866025) / sin(30 deg) = 0.133975 / 0.5 ~ 0.267949.
What are the half-angle identities?
The half-angle identities are a set of trigonometric formulas that express sin(theta/2), cos(theta/2), tan(theta/2), and their reciprocals in terms of the cosine (and sometimes sine) of the full angle theta. They follow directly from the double-angle identities: if you replace x with theta/2 in cos(2x) = 1 - 2sin^2(x) and cos(2x) = 2cos^2(x) - 1 and solve for the single-angle functions, you get the half-angle formulas. These identities are essential for simplifying integrals, solving triangle problems, and evaluating exact values of trig functions at non-standard angles such as 15, 22.5, and 75 degrees.
How to determine the correct sign
Every half-angle formula for sine and cosine carries a plus-or-minus (+-) sign, because the square root is always positive but the trig function can be negative. The correct sign depends on which quadrant the half angle theta/2 falls into. For sine, the result is positive in quadrants 1 and 2 (where 0 < theta/2 < 180 deg), and negative in quadrants 3 and 4 (where 180 deg < theta/2 < 360 deg). For cosine, the result is positive in quadrants 1 and 4, and negative in quadrants 2 and 3. The two alternate tangent formulas -- (1 - cos theta) / sin theta and sin theta / (1 + cos theta) -- do not carry a +/- sign at all, because the signs of the numerator and denominator automatically produce the correct result for every quadrant.
Practical applications
Half-angle identities appear in calculus (simplifying integrals of powers of sine and cosine), analytic geometry (converting between Cartesian and polar or parametric forms), signal processing (Weierstrass substitution, which maps a rational function of sin and cos to a rational function of a single variable), and exact geometry (computing the diagonal of a regular polygon or the half-apex angle of a cone). In navigation and surveying, the tangent half-angle formula is used in the Mollweide equations for solving oblique triangles.
Half-angle vs double-angle identities
Double-angle identities (sin(2x) = 2 sin x cos x, cos(2x) = cos^2 x - sin^2 x) express a function of 2x in terms of functions of x. Half-angle identities do the reverse: they express a function of x/2 in terms of functions of x. Together they let you move fluidly between multiples and fractions of an angle. For instance, if you know cos(60 deg) = 0.5 exactly, you can derive sin(30 deg), cos(30 deg), and all the other values at 30 deg without a calculator, and then repeat the process to find exact values at 15, 7.5, or any other binary fraction of 60 deg.
Half-angle identity quick-reference
| Function | Half-angle identity | Alternate form |
|---|---|---|
| sin(theta/2) | +/- sqrt[(1 - cos theta) / 2] | -- |
| cos(theta/2) | +/- sqrt[(1 + cos theta) / 2] | -- |
| tan(theta/2) | +/- sqrt[(1 - cos theta) / (1 + cos theta)] | (1 - cos theta) / sin theta or sin theta / (1 + cos theta) |
| csc(theta/2) | +/- sqrt[2 / (1 - cos theta)] | 1 / sin(theta/2) |
| sec(theta/2) | +/- sqrt[2 / (1 + cos theta)] | 1 / cos(theta/2) |
| cot(theta/2) | +/- sqrt[(1 + cos theta) / (1 - cos theta)] | (1 + cos theta) / sin theta |
All six half-angle identities. The sign (+/-) is chosen based on the quadrant of theta/2.
Frequently asked questions
How do I find sin of a half angle?
Use the identity sin(theta/2) = +/- sqrt[(1 - cos theta) / 2]. First compute cos theta, then substitute into the formula. Determine the sign by checking which quadrant theta/2 falls into: positive if theta/2 is in Q1 or Q2 (between 0 and 180 deg), negative if in Q3 or Q4.
Why does the tangent half-angle formula have two alternate forms?
Starting from tan(theta/2) = sin(theta/2) / cos(theta/2) and substituting the sqrt forms of each, you can rationalise the expression in two ways. Multiplying numerator and denominator by (1 + cos theta) yields sin theta / (1 + cos theta). Multiplying by (1 - cos theta) yields (1 - cos theta) / sin theta. Both are equivalent and neither carries a +/- sign, which makes them safer to use than the sqrt form when you are working by hand.
Can I use this for angles in radians?
Yes. Select "Radians" from the unit dropdown and enter your angle in radians. The calculator converts internally and determines the quadrant of theta/2 in degrees for sign resolution, so the results are exactly the same as the degree mode for the same geometric angle.
What is the half angle of 90 degrees?
The half angle is 45 degrees. sin(45 deg) = cos(45 deg) = sqrt(2)/2 ~ 0.707107, and tan(45 deg) = 1. You can verify this with the identity: cos(90 deg) = 0, so sin(45 deg) = sqrt[(1 - 0)/2] = sqrt(0.5) = sqrt(2)/2.
Is cos(theta/2) the same as cos(theta)/2?
No. cos(theta/2) is the cosine of half the angle, while cos(theta)/2 is half the cosine of the full angle. For example, with theta = 60 deg: cos(30 deg) ~ 0.866, but cos(60 deg)/2 = 0.5/2 = 0.25. The two are equal only at theta = 0.
How do I find exact values for 15 degrees and 22.5 degrees?
For 15 deg, apply the half-angle formulas with theta = 30 deg: cos(30 deg) = sqrt(3)/2, so sin(15 deg) = sqrt[(1 - sqrt(3)/2)/2] = sqrt[(2 - sqrt(3))/4] = sqrt(2 - sqrt(3))/2. For 22.5 deg, use theta = 45 deg: cos(45 deg) = sqrt(2)/2, so sin(22.5 deg) = sqrt[(1 - sqrt(2)/2)/2] = sqrt[(2 - sqrt(2))/4].