Half Angle Calculator

sin(θ/2) = √((1 − cos θ) / 2) cos(θ/2) = √((1 + cos θ) / 2) tan(θ/2) = (1 − cos θ) / sin θ = sin θ / (1 + cos θ)

θ

The angle you start with, in degrees or radians. It can be negative and it can be past a full turn, and it is brought into range before anything is halved

θ/2

The half angle, always reported in the range from zero up to a half turn. Every other number on the page is read here rather than at the angle you typed

cos θ

The cosine of the whole angle. It is the only input the sine and cosine identities need — both roots are built out of it, which is why a reading at an awkward angle can still be exact

sin θ

The sine of the whole angle, used by the two fractional forms of the tangent. It is the divisor, so it must not be zero — and the angles where it is zero are exactly the angles where the half runs into a right angle

Reach for this page when the angle is going to be halved and you want the three ratios at the half without working the identities out by hand — an oscillator at half the driving frequency, an angle between two mirror surfaces, a chord seen from half the central angle, a step that is half of a full turn or a quarter of one. The formulas below the fold are the same three identities the calculator applies, so the page doubles as a check on a hand calculation. Two neighbours cover the rest. If what you have is the angle rather than its half, the double angle formulas go the other way, and the two pages agree everywhere both are defined. And if you want the three ratios at the angle in the box rather than at its half, the trigonometry page reads them there and nothing is halved.

1. An angle of 30 degrees, halving to 15

   1. Half of thirty degrees is fifteen degrees, so everything is read there
   2. The identity for the sine: the cosine of thirty is 0.866, so one minus that is 0.134, and half of 0.134 is 0.067, whose square root is 0.2588
   3. The cosine is the same root with a plus inside: one plus 0.866 is 1.866, half of that is 0.933, and its square root is 0.9659
   4. The tangent is their ratio, 0.2588 divided by 0.9659, which is 0.2679 — and the fractional form agrees: 0.134 divided by 0.5 is 0.268

   The angle the page loads with, and the one that shows the mistake the page is aimed at. The sine of thirty degrees is 0.5, so half of that would be 0.25 — but the sine of fifteen degrees is 0.2588, a little more than half. The two readings are close enough that the wrong answer looks plausible, which is exactly why the page prints it. Notice also that the whole angle has an exact cosine here, a half of root three, and the half angle came out irrational anyway: halving an angle takes you off the chart rather than onto it.

2. An angle of 90 degrees, halving to 45

   1. Half of ninety degrees is forty-five
   2. The cosine of ninety is zero, so both identities reduce to the square root of a half, which is 0.7071
   3. The two roots are the same number here because the thing inside them is the same: one minus zero and one plus zero
   4. Equal readings mean the tangent is exactly one, with no rounding

   The one row where a famous angle comes out of an angle that is not famous for this. Forty-five degrees is one of the few angles whose three ratios can be written down exactly — the sine and cosine are both a half of root two, and the tangent is one. The route to it is a half angle rather than a memorised chart, and this row is that route. It is also the row where the sine and the cosine are equal, which happens at forty-five and at two hundred and twenty-five and nowhere else on the table.

3. An angle of 400 degrees, which is the same as 40

   1. Four hundred degrees is forty degrees plus one whole turn
   2. One whole turn changes nothing about where the angle points, so the reading is taken at forty
   3. Half of forty is twenty, so the three ratios are those of twenty degrees: 0.342, 0.9397 and 0.364

   Enter forty degrees into the same box and every one of the four numbers is identical — that is the whole point of this row. Halving before normalising would give two hundred degrees here and twenty there, two different answers to the same question, which is why the order is fixed the other way round. The same holds for seven hundred and fifty degrees, which halves to the reading of a fifteen degree angle.

4. An angle of minus 30 degrees, halving to 165

   1. Minus thirty degrees is three hundred and thirty degrees, a third of a turn short of a full one
   2. Half of three hundred and thirty is one hundred and sixty-five
   3. One hundred and sixty-five is in the second quadrant, so its sine is positive and its cosine is negative
   4. The tangent is therefore negative, minus 0.2679

   The row that catches people out. A negative angle does not halve to a negative half, because the angle is brought into range first; half of a negative angle is an angle in the upper half of the circle. The sine here is the same 0.2588 as the thirty degree row, and that is not a coincidence — fifteen and one hundred and sixty-five are mirror images across the vertical axis, so they share a sine and their cosines are opposites. The tangent flips sign along with the cosine.

Three things this page does not do. First, the tangent is refused at one hundred and eighty degrees, at five hundred and forty, at minus one hundred and eighty, and at every angle those represent plus or minus a whole turn, because the half of each is a right angle and the tangent has no value there. The refusal takes the whole reading with it even though the sine and the cosine at the half are ordinary numbers — one and zero. Second, the three ratios are reported to four decimal places and most of them are irrational; the printed half angle is a different matter, since halving a whole number of degrees gives at worst a half degree and two decimal places are enough to hold it exactly. Third, the page cannot be run backwards. The half of an angle is shared by everything a full turn away from it and by nothing else, so a reading of twenty degrees tells you the original was forty, or four hundred, or any of the infinitely many angles between them — and a half of one hundred and ninety five degrees would come from an original outside the range the box accepts, so that direction has no answer here at all.

Is the sine of a half angle half the sine of the angle?

No, and that is the mistake this page is built around. The sine of thirty degrees is 0.5, so half of that is 0.25 — but the sine of fifteen degrees is 0.2588. The two are close enough that the wrong answer looks reasonable, which is why the page prints the right one with the steps. What the sine of a half angle actually is, is the square root of one minus the cosine of the whole angle, over two: a root of a cosine, not a fraction of a sine.

What are the three half angle identities?

The sine of a half angle is the square root of one minus the cosine of the whole angle, over two. The cosine of a half angle is the same root with a plus inside instead of a minus. The tangent of a half angle can be written three ways — one minus the cosine over the sine, the sine over one plus the cosine, or the square root of one minus the cosine over one plus the cosine — and all three give the same number. The fractional forms are usually the ones to reach for, since they avoid taking a square root at all.

Why does a negative angle not halve to a negative angle?

Because the angle is brought into range before it is halved, not after. Minus thirty degrees is three hundred and thirty degrees, and half of three hundred and thirty is one hundred and sixty-five. Halving first would give minus fifteen, which is a real angle, but it is not in the range the output reports and it would disagree with the rest of the page about which of the two mirror-image angles is meant. The reading is always the one in the upper half of the circle.

Why does the page refuse 180 degrees?

Because half of one hundred and eighty is ninety, and the tangent of ninety degrees has no value. The sine and the cosine at ninety are fine — one and zero — but the page reports the tangent along with them, and a partial answer presented as a whole one is worse than none. The same refusal covers five hundred and forty degrees and minus one hundred and eighty, which are the same angle as one hundred and eighty, and every angle a whole turn away from those.

How is this different from the double angle formulas?

They are the same three relations read in opposite directions. The double angle formulas start from an angle and give the sine, cosine and tangent at twice it; the half angle formula starts from an angle and gives the three ratios at half of it. The difference that matters is where the rounding goes. Doubling keeps you on whole degrees, so the reading at twice thirty is exactly sixty; halving often takes you off the chart, so the half of thirty is fifteen and its readings are irrational.

Can I enter the angle in radians?

Yes — there is a unit selector beside the box, and the half angle is reported in degrees either way, since that is the unit the output declares. A sixth of pi radians and thirty degrees are the same angle and give the same four numbers. The one thing to watch is that the unit applies to the angle you type and not to the half, which is always shown in degrees.

• Half-Angle Formulas — the three identities this page applies, with the sign questions the page resolves by keeping the half in the upper half of the circle — Wolfram MathWorld (United States)
• Double-Angle Formulas — the same relations read the other way, and the page these identities are usually met from first — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions of sine, cosine and tangent, and where each of them has no value — Wolfram MathWorld (United States)
• Sine, Cosine and Tangent — the ratios in a right triangle, from first principles — Math is Fun (United Kingdom)