Double Angle Calculator

sin 2θ = 2 sin θ cos θ cos 2θ = cos²θ − sin²θ tan 2θ = 2 tan θ ÷ (1 − tan²θ)

θ

The angle you start with, in degrees or radians. It can be negative and it can be past a full turn, and what the page doubles is this angle, never the number it produces

2θ

The doubled angle. It is reported first, in degrees, because everything else on the page is read at that angle rather than at the one you typed

sin θ, cos θ

The sine and the cosine of the original angle. The identity multiplies them together rather than doubling either one, which is why the answer is not twice anything

Reach for this page when the angle is going to be doubled and you want to know where it lands without working the identities out by hand — a rotation applied twice, a phase that advances by twice the step, a wavelength or a period read at twice the frequency. The double angle formulas below the fold are the same three identities the calculator applies, so the page doubles as a check on a hand calculation. If what you have is the doubled reading and you want the original angle back, that is the half angle direction and it is not a function this page performs. And if you want the three ratios at the angle you typed rather than at its double, the trigonometry page does exactly that and the two agree wherever both are defined.

1. An angle of 30 degrees, doubling to 60

   1. Twice thirty degrees is sixty degrees, so everything is read there
   2. sin 60 is √3/2, about 0.866, and cos 60 is a half
   3. The tangent is their ratio, √3, about 1.7321

   The angle the page loads with, and the one that shows why the page exists. The sine of thirty degrees is 0.5, so twice that would be 1 — but the sine of sixty degrees is 0.866, and no amount of doubling the first reading produces the second. The identity explains where 0.866 comes from: two times 0.5 times 0.866, the product of the sine and the cosine of thirty. The doubled angle is bigger than the original by a factor of two; the ratio is not.

2. An angle of 22.5 degrees, doubling to 45

   1. Twice twenty-two and a half is forty-five degrees
   2. At forty-five degrees the point is on the line y = x, so the sine and the cosine are equal
   3. Equal readings mean the tangent is exactly one, with no rounding

   Twenty-two and a half is half of forty-five, which makes this row the half angle formula read in the direction it is usually wanted. Forty-five is one of the few angles whose three ratios are exact, and it is out of reach of the usual chart, so the route to it is a doubled value rather than a memorised one. Both ratios come out 0.7071 because at that angle the two coordinates are equal, and the tangent is one exactly rather than 0.9999.

3. An angle of 90 degrees, doubling to 180

   1. Twice ninety is one hundred and eighty degrees, half a turn
   2. The point is on the negative x-axis, so the cosine is minus one and the sine is zero
   3. The tangent is zero divided by minus one, which is zero

   The one row where the doubled angle reaches an extreme: the cosine is at its smallest possible value, minus one. Nothing on this page fails at ninety degrees, even though ninety is the angle the trigonometry page refuses — the tangent is not being read there, it is being read at one hundred and eighty, where it is perfectly well behaved and equal to zero. Moving the reading to a different angle is enough to move it off a pole.

4. An angle past a full turn, 750 degrees

   1. Seven hundred and fifty degrees is thirty degrees plus two whole turns
   2. Doubling it gives fifteen hundred degrees, which is sixty degrees plus four whole turns
   3. Four whole turns change nothing, so the reading is the same as at sixty

   The four columns are identical to the thirty degree row, and they should be: seven hundred and fifty and thirty are the same angle, so they have the same double and the same three ratios there. Nothing here treats a large angle as a special case — the doubled angle is brought back into range before it is read, and whole turns are discarded. Five degrees under a million behaves the same way for the same reason.

Three things this page does not do. First, the tangent of a doubled angle is refused at forty-five, at one hundred and thirty-five, and at every angle that doubles to a right angle or three-quarters of a turn — including the negative and past-a-turn versions of those. The refusal takes the whole angle with it, even though the sine and the cosine of the doubled angle are ordinary numbers there. Second, the readings are given to four decimal places, and most of them are irrational: the exact sine of a doubled thirty degrees is root three over two, and 0.866 is where that lands after rounding. The cosine is worse behaved than the sine here — the identity subtracts two squares, and the two are nearly equal near forty-five degrees, so the small answer that comes out is the difference of two larger numbers. Third, the page does not invert. Given a sine of 0.866 for a doubled angle, that doubled angle could be sixty or one hundred and twenty, and the original could be any of the infinitely many angles that double to either — recovering one angle from a ratio needs a decision this page does not make.

Is the sine of a doubled angle twice the sine of the angle?

No, and this is the mistake the page is built around. The sine of thirty degrees is 0.5, but the sine of sixty degrees is 0.866, not 1. Doubling an angle is not a scaling of its ratios. What the doubled angle's sine actually is, is twice the product of the original angle's sine and cosine — a multiplication of two different readings rather than a doubling of one.

Why does the page refuse 45 degrees?

Because forty-five doubles to 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 as well, and a partial answer presented as a whole one is worse than none. The same refusal covers one hundred and thirty-five degrees, which doubles to two hundred and seventy, and every angle a half turn from either.

What are the three double angle identities?

The sine of a doubled angle is twice the product of the sine and the cosine of the original. The cosine of a doubled angle is the square of the cosine minus the square of the sine, which can also be written as one minus twice the sine squared or as twice the cosine squared minus one — all three are the same number, and the form to use is the one whose original ratio you already have. The tangent of a doubled angle is twice the original tangent divided by one minus the square of that tangent.

Why is 22.5 degrees worth knowing?

Because its double is forty-five, and forty-five is one of the handful of angles with exact ratios — a half and a half for the sine and the cosine, and exactly one for the tangent. Twenty-two and a half is not on any standard chart, but it is half of one that is, so it is reachable by reading a double angle formula backwards. That backwards reading is what the half angle formulas are.

Can the angle be negative or larger than a full turn?

Yes, and both are handled the same way: the angle is doubled and the result is brought back into range, so seven hundred and fifty degrees reads exactly like thirty and minus thirty reads exactly like three hundred and thirty. Whole turns are discarded because they change nothing about where the doubled angle points. The only thing to watch is that the reading is always reported in the range from zero up to a full turn.

Can I enter the angle in radians?

Yes — there is a unit selector beside the box, and the doubled angle is reported in degrees either way, since that is the unit the output declares. A third of pi radians and sixty degrees are the same angle and give the same four numbers. As always, check which unit a source you are comparing against assumes, because most published tables mean degrees.

• Double-Angle Formulas — the three identities this page applies, and how they follow from the addition formulas — Wolfram MathWorld (United States)
• Trigonometric Addition Formulas — the identities the double angle ones are a special case of, with everything derived rather than asserted — 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)