Half Angle Calculator

Half an angle, without halving the sine

Halving an angle is easy. Halving its sine is not. The sine of 60 degrees is 0.8660 and the sine of 30 degrees is 0.5, which is nowhere near half of 0.8660. The half-angle identities exist to close that gap: they rebuild the sine, cosine and tangent of theta/2 out of the cosine of theta, using one square root and nothing else.

That matters whenever a drawing, a saw scale or a survey line gives you an angle but the number you need belongs to half of it. A 90 degree corner on a picture frame is cut at 45. A 120 degree bend in a handrail is mitred at 60. A chamfer callout of 30 degrees means the tool sits at 15 from the surface. In each case the angle you can measure and the angle you must set are not the same number, and guessing that the trig values split in the same ratio is how parts come out oversized.

The calculator above takes one angle, divides it in half, evaluates sine, cosine and tangent at the half angle, and reports the sign it used. The sign is the part that trips people up, so it gets its own section below.

Read the 210 degree row twice. The half angle, 105 degrees, sits in the second quadrant, so its cosine is negative and its tangent is negative while its sine stays positive. A calculator that drops the sign will still return a number, and the number will be wrong by a minus sign that is easy to miss on a drawing.

The three identities and the sign in front of them

Every half-angle formula has the same shape: a square root of a ratio built from cos theta, with a plus or minus in front. The three standard forms are below, written the way they appear in the NIST Digital Library of Mathematical Functions and in any precalculus text.

The two tangent forms without a plus or minus are worth memorising. Because they divide one signed value by another signed value, the sign comes out correct on its own, with no quadrant reasoning required. The catch is the denominator: sin theta / (1 + cos theta) blows up when theta is 180 degrees, and (1 - cos theta) / sin theta blows up at 0 and 180 degrees. For those two angles, tangent of the half angle is undefined anyway, so the calculator reports it as such instead of printing a huge number.

The square-root forms look like they only need the cosine, and they do, but the cosine alone cannot tell you whether theta sits above or below the horizontal axis. That is exactly why the plus or minus is there. The next section pins the sign down.

Which sign to use: read the quadrant of the half angle

The sign belongs to the half angle, not to theta. Work out where theta/2 lands on the circle, then apply the usual rule for that quadrant: sine is positive in the first and second, cosine is positive in the first and fourth, tangent is positive in the first and third.

For theta between 0 and 360 degrees the half angle runs from 0 to 180, so the sine of the half angle is never negative and the cosine of the half angle changes sign at theta = 180. That single fact covers most school and shop problems.

Angles outside 0 to 360 are handled by reducing first. Add or subtract 360 until the angle lands in the base range, then halve. The trig values repeat every 360 degrees, so sin(750/2) is the same as sin(30/2) once 750 is reduced to 30. Negative angles work the same way: -60 degrees reduces to 300, whose half angle is 150, so the sine of the half angle is positive and the cosine is negative. If your answer disagrees with the calculator on a negative input, check whether you halved before or after reducing.

Worked example: half of 60 degrees

Take theta = 60 degrees, where cos 60 = 0.5 and sin 60 = 0.8660. The half angle is 30 degrees, and the identities should reproduce the values every student memorises.

The results are sin 30 = 0.5, cos 30 = 0.8660 and tan 30 = 0.5774, which match the table values exactly. Both square roots were taken as positive because the half angle, 30 degrees, sits in the first quadrant. Nothing about the input 60 degrees was needed beyond its cosine and sine.

Notice how the two square roots trade places. At theta = 60 the sine of the half angle equals the cosine of theta, and the cosine of the half angle equals the sine of theta. That swap is a property of complementary angles, and it is a fast sanity check on any result near 90 degrees.

Worked example: half of 210 degrees, where the cosine turns negative

Now take theta = 210 degrees. Its cosine is -0.8660 and its sine is -0.5. The half angle is 105 degrees, in the second quadrant, so the sine of the half angle is positive, the cosine is negative and the tangent is negative.

The tangent check is the useful one. The sign-free form, sin theta / (1 + cos theta), gives -0.5 divided by 0.1340, which is -3.7313. The exact value of tan 105 is -3.7321, and the small gap is only the rounding in the intermediate 0.1340. No quadrant rule was applied to get the sign, yet the answer came out negative, which is what the second quadrant demands.

If you had used the square-root form and taken the positive root out of habit, you would have reported cos 105 as 0.2588 and tan 105 as 3.7321. Both are wrong by a sign, and a sign error in a bearing or a rotation matrix sends the result to the wrong side of the circle.

The tangent half-angle substitution

Setting t = tan(theta/2) turns every trigonometric expression into algebra in t. The three relations that make the trick work are sin theta = 2t / (1 + t squared), cos theta = (1 - t squared) / (1 + t squared), and d theta = 2 dt / (1 + t squared). The NIST DLMF lists them as equation 4.21.36, and calculus courses use them to integrate rational functions of sine and cosine.

Check the substitution on theta = 60 degrees, where t = tan 30 = 0.5774. The sine becomes 2 x 0.5774 / (1 + 0.3333), which is 1.1547 / 1.3333 = 0.8660, the sine of 60. The cosine becomes (1 - 0.3333) / 1.3333, which is 0.6667 / 1.3333 = 0.5, the cosine of 60. Both land where they should.

This is also the form to reach for when a problem gives you a tangent and asks for the angle. Invert it once, theta = 2 arctan t, and the arctangent returns a value between -90 and 90 degrees, so theta lands between -180 and 180. Add 360 if your context needs a positive bearing.

Exact values you get by halving a known angle

Halving an angle you already know turns a familiar value into a new exact one. The decimals below come straight from the square-root forms, and the radical column is what a textbook answer key would print.

The 15 and 75 degree rows are the payoff. Both come from halving 30 and 150 degrees, and both are exact values built from square roots of 2 and 3, which is why machinists and drafters still use them when a 30 degree chamfer has to be split. The 22.5 degree row comes from halving 45, and the nested radical is the reason that value shows up in octagon geometry, where each interior angle is 135 degrees and each miter is 67.5.

Where half angles show up, and the mistakes to avoid

Woodworking is the most common place. A miter saw scale is marked in the angle from the fence, not the angle of the corner, so a 90 degree corner needs the saw set to 45 and a 120 degree corner needs 60. Crown molding adds a second tilt on top of the miter, and the two settings are both half-angle calculations against the spring angle of the profile.

Machining uses the same idea for chamfers and countersinks. A callout of 82 degrees included angle on a flat-head screw means the tool cuts 41 degrees per side. Roof pitch, ramp slope and stair stringer angles all get quoted as a rise and run that has to be halved before it becomes a cut. In graphics, rotating a shape by half of a symmetry angle is how a repeated motif lines up.

Four mistakes account for most wrong answers:

• Halving the trig value instead of the angle. sin(60)/2 is 0.4330, while sin(30) is 0.5. They are different numbers and only one is correct.
• Dropping the sign. A positive square root is right only when the half angle is in the first or second quadrant.
• Mixing degrees and radians. Half of 60 radians is 30 radians, which is a different point on the circle from 30 degrees. Set the calculator unit before typing.
• Using the tangent form at the wrong angle. Both sign-free forms divide by a term that reaches zero at 180 degrees, and the square-root form does the same at 90 degrees.

For the reverse direction, doubling an angle back up, the double angle calculator applies sin 2 theta, cos 2 theta and tan 2 theta from the same starting values. If you only need a reference angle or a coterminal angle rather than a half angle, the reference angle calculator and the coterminal angle calculator cover those cases.

The formulas themselves are catalogued in section 4.21 of the NIST Digital Library of Mathematical Functions, and the worked derivations appear in the OpenStax precalculus chapter on double-angle and half-angle formulas. MathWorld's half-angle entry collects the same identities with the sign conventions spelled out.