2θ is 60° and θ/2 is 15°. The half-angle signs here are sine +, cosine +.
Double angle: 60°
Sine, cosine and tangent of the doubled angle with the identity each comes from
Function
Identity
Value
sin 2θ
2 sin θ cos θ
0.8660254
cos 2θ
cos²θ − sin²θ
0.5
2cos²θ − 1
0.5
1 − 2sin²θ
0.5
tan 2θ
2 tan θ ÷ (1 − tan²θ)
1.7320508
All three forms of cos 2θ give the same number, as they must. Which one is convenient depends on what you already know — and the 2cos²θ − 1 form is the one that rearranges into the half-angle formula.
sin 15°
0.25881905
the half angle
cos 15°
0.96592583
the half angle
tan 15°
0.26794919
the half angle
Half-angle signs
sine +, cosine +
decided by the quadrant of θ/2
The double-angle formulas are the sum formulas with both angles the same: sin(θ+θ) and cos(θ+θ). Nothing new is being asserted, which is why they are worth deriving once rather than memorising.
cos 2θ has three equivalent forms, and all three are evaluated here. Which one is convenient depends on what you already know — the 2cos²θ − 1 form is the one that rearranges into the half-angle formula.
The half-angle formulas carry a ±, and the sign is decided by the quadrant of θ/2 rather than of θ. Here θ/2 is 15°, which puts sine positive and cosine positive.
That ± is the part most often dropped. It is not decoration: for θ = 300°, θ/2 = 150° and the half-angle sine is positive while sin 300° is negative.
The half-angle ± is resolved from the quadrant of θ/2 rather than of θ, which is where it usually goes wrong.
What this tool shows
cos 2θ has three equivalent forms, and the half-angle formulas carry a ± that a table cannot resolve. This page evaluates all three forms and settles the sign from the quadrant of θ/2 — which is the step that is usually skipped and usually wrong.
Sine, cosine and tangent of 2θ
All three equivalent forms of cos 2θ
Sine, cosine and tangent of θ/2
The sign each half-angle value takes
Where the formulas come from
The power-reducing rearrangement
2θ and θ/2 together All three cos forms The ± settled Derived, not asserted
Updated 7 September 2026 · Works in any browser, no installation
sin 2θ = 2 sin θ cos θ, and cos 2θ is any of three equivalent expressions. The half-angle formulas are those rearranged, and they carry a ± that the formula alone cannot settle — the quadrant of θ/2 decides it, which is the step most workings skip.
At a glance
Formula shown
sin 2\u03b8 = 2 sin \u03b8 cos \u03b8. cos 2\u03b8 = cos\u00b2\u03b8 \u2212 sin\u00b2\u03b8 = 2cos\u00b2\u03b8 \u2212 1 = 1 \u2212 2sin\u00b2\u03b8. tan 2\u03b8 = 2 tan \u03b8 / (1 \u2212 tan\u00b2\u03b8). sin(\u03b8/2) = \u00b1\u221a((1 \u2212 cos \u03b8)/2) and cos(\u03b8/2) = \u00b1\u221a((1 + cos \u03b8)/2), with the sign set by the quadrant of \u03b8/2.
Scenario support
Simplifying an expression before integrating it; finding an exact value for 15\u00b0 or 22.5\u00b0; rewriting a squared trig term so it can be integrated.
Educational estimate
Planning support from the values you enter — not professional advice.
Where the formulas come from
Neither family needs memorising if you know the sum formulas, because both are those with the two angles set equal.
sin(A + B) = sin A cos B + cos A sin B. Put B = A and it becomes sin 2A = 2 sin A cos A. That is the whole derivation.
cos(A + B) = cos A cos B − sin A sin B. Put B = A and it becomes cos 2A = cos²A − sin²A.
The half-angle formulas come from rearranging the cos 2θ identity. Writing cos 2θ = 1 − 2sin²θ, solving for sin θ, and then replacing θ with θ/2 gives sin(θ/2) = ±√((1 − cos θ)/2).
Three families, one derivation, and the ± appears exactly at the point where a square root is taken.
Three forms of cos 2θ
cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ. All three are the same number, and the page evaluates all three so you can see it.
They differ only in what they need. The first is the raw derivation. The second is what to use when you know the cosine. The third is what to use when you know the sine.
The Pythagorean identity converts between them: substituting sin²θ = 1 − cos²θ into the first gives the second, and the other substitution gives the third.
The second form is the one that matters most, because rearranging it is where the half-angle cosine formula comes from — and where the power-reducing formula comes from too.
Where the ± comes from
The half-angle formulas produce a square root, and a square root has two signs. The formula gives the size; something else has to give the sign.
That something is the quadrant of θ/2, not of θ. This is the point that gets dropped, and dropping it produces an answer with the right magnitude and the wrong sign.
A concrete case. Take θ = 300°, which is in the fourth quadrant where sine is negative. Then θ/2 = 150°, which is in the second quadrant where sine is positive. So sin(150°) is positive even though sin(300°) is negative, and reading the sign off θ would get it backwards.
The page computes θ/2, finds its quadrant, and applies the sign from there. It reports which signs it used, so the working can be checked rather than trusted.
Power reduction
Rearranging cos 2θ = 2cos²θ − 1 for cos²θ gives cos²θ = (1 + cos 2θ)/2. Similarly sin²θ = (1 − cos 2θ)/2.
These are the power-reducing formulas, and they matter because a squared trigonometric term cannot be integrated directly while a plain cosine can. Rewriting sin²x as (1 − cos 2x)/2 turns an impossible integral into two easy ones.
The same trick handles higher powers by applying it repeatedly, and it is why the double-angle identity turns up constantly in calculus rather than only in geometry.
It is the same algebra as the half-angle formulas, before the square root is taken — which is why this page treats them as one topic rather than three.
Where they get used
Exact values for awkward angles. 15° is half of 30°, so the half-angle formula gives sin 15° exactly — √(2 − √3)/2, which no unit-circle chart contains. 22.5° comes from 45° the same way.
Integration. Power reduction, above. It is the standard first move on any integral containing a squared sine or cosine.
Physics and signal work. Squaring a sine wave — which is what computing power from a current does — produces a term at twice the frequency plus a constant. That is the double-angle identity, and it is why AC power has a component at 100 Hz on a 50 Hz supply.
Simplifying before differentiating. An expression rewritten with a double angle is often much easier to differentiate than the product it came from.
The triple angle, and why it stops
There are triple-angle formulas: sin 3θ = 3 sin θ − 4 sin³θ, and cos 3θ = 4 cos³θ − 3 cos θ.
They follow from the sum formulas the same way, by writing 3θ as 2θ + θ. And the pattern continues: every multiple angle has a formula, given by the Chebyshev polynomials.
They stop being taught after three because they stop being useful. The degree grows with the multiple, the expressions get long, and in practice the double and half angle cases cover almost everything — which is why this page covers those two and points at the general pattern rather than tabulating it.
Sources and methodology
The identities are standard; these are the references for them and for the power-reducing use.
Method. The double-angle values are computed directly from 2θ as well as through each identity, and the suite requires all of them to agree — including all three forms of cos 2θ, which would diverge instantly on a sign error. The half-angle sign is decided from the quadrant of θ/2 rather than of θ, and the suite checks that decision at every ten degrees through two full turns. That engine is verified on every change against 71 hand-written assertions, including that all three forms of cos 2θ match a direct computation across four hundred and eighty angles, and that the half-angle sine sign follows the quadrant of θ/2 at every one of seventy-two test angles. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Sum and Difference IdentitiesEach identity evaluated twice — through the formula and by computing the combined angle directly — so the agreement between them is evidence rather than a restatement.
Trigonometric FunctionsAll six functions at once with exact surd values at the sixteen special angles — and tan 90° reported as undefined rather than as the 1.6 × 10¹⁶ a double returns.
Unit CircleWhere an angle lands, with exact surd coordinates and the radian measure as a multiple of π — because the coordinates ARE the cosine and sine.
Inverse Trigonometric FunctionsThe principal value AND the general solution, because a calculator hands you one member of an infinite family and the context decides which one your problem wants.
Phase ShiftAmplitude, period, phase shift and midline for y = A sin(Bx + C) + D — with the shift computed as −C/B, which is the step everyone skips.
Special Right Triangles30-60-90 and 45-45-90 from any one side, in exact surd form as well as decimals — with the warning that the long leg is √3 times the short, not twice.
These come from the sum and difference identities — set B = A and the double-angle formulas fall out — and the Sum and Difference Identities Calculator shows that step with both sides evaluated.
Educational use disclaimer
This is an educational tool. The values are exact to double precision; the ± resolution assumes the half angle you want is θ/2 itself rather than a coterminal alternative.
Published double and half angle on one page, because the half-angle formula IS the double-angle one rearranged and splitting them would duplicate the derivation.
All three forms of cos 2θ are evaluated side by side, since which one is convenient depends on what you already know and the 2cos²θ − 1 form is the one that becomes the half-angle formula.
The ± in the half-angle formulas is resolved from the quadrant of θ/2 rather than of θ, which is the part most often dropped and the part that is actually wrong when it is.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.