How this instrument works
sin(2θ) = 2sinθcosθ looks like a standalone rule to memorize, but it isn't one — it falls straight out of angle-addition, sin(x+y) = sinx cosy + cosx siny, the instant one angle is set equal to the other. Two cross terms, sinθcosθ and cosθsinθ, turn out to be one product written in reversed order, so instead of staying separate they simply add: sinθcosθ + cosθsinθ = 2sinθcosθ. This sheet computes both sides on one angle and reports they never disagree — that agreement is the entire content of the identity, laid bare rather than asserted.
One tidy reason explains why doubling comes out exact rather than approximate. Treat an angle as a complex exponential: e^(i·2θ) equals (e^(iθ))², and squaring (cosθ + i·sinθ) gives cos²θ − sin²θ as its real part and 2sinθcosθ as its imaginary part. Match that imaginary part against sin(2θ) and out falls double-angle — a byproduct of squaring one rotation, since doubling an angle is literally squaring its representative point on a circle of unit radius.
An edge case worth knowing sits inside Difference itself. viaDoubleAngle and viaAddition both build from one identical floating-point product of sinθ and cosθ, and doubling that floating-point number by adding it to itself never rounds — it only shifts a binary exponent — so subtraction returns exactly 0.0 rather than a near-zero residue at every angle tried, including 0° and 90°, where sine itself vanishes or peaks.
- Enter your angle into Angle, θ; default unit is degrees, with radians and turns also available.
- Read sin(2θ) via double-angle formula for a fast result: 2sinθcosθ evaluated directly at your angle.
- Read sin(θ+θ) via the addition formula alongside it — same angle, run through a general sum rule rather than a shortcut.
- Check Difference (always 0): it subtracts both results and should hold at 0.0 no matter what angle you entered.
- Change your angle and watch all three outputs update together; equality never breaks, which is this sheet's whole point.
Worked example — θ = 40°
Set Angle, θ to 40°, which this sheet stores internally as 0.6981317007977318 radians. sin(2θ) via double-angle formula computes 2 × sin(40°) × cos(40°) = 2 × 0.64278761 × 0.76604444 = 0.9848077530122079 — that is sin(80°), reached by a fast route: one multiplication and one doubling.
sin(θ+θ) via the addition formula takes a longer route on that identical angle: sin(40°)cos(40°) + cos(40°)sin(40°) = 0.49240387650610395 + 0.49240387650610395 = 0.9848077530122079, matching to all seventeen figures. Difference (always 0) reports exactly 0.0, so a shortcut and a general sum rule turn out to be one statement, not two separate facts that happen to agree.
Questions
Where does the double-angle formula for sine actually come from?
It comes from angle-addition, sin(x+y) = sinx cosy + cosx siny. Setting y equal to x makes both terms identical, sinθcosθ and cosθsinθ, so they combine into 2sinθcosθ rather than needing separate reasoning. This sheet runs both versions on one θ to show they never disagree.
How is this different from a page that doubles sin, cos, and tan together?
This one deliberately computes sin(2θ) two independent ways and subtracts those results, instead of reporting sinθ, cosθ, and tanθ doubled together for convenience — that combined lookup lives on this site's separate double-angle-formula page. Here a proof is the point, not a triple listing.
Why does Difference read exactly 0.0 instead of something merely close to zero?
Both routes multiply one identical pair of numbers, sinθ and cosθ, and IEEE 754 arithmetic gives one identical result regardless of multiplication order. Doubling a floating-point value by adding it to itself is also exact — it only shifts a binary exponent — so this identity holds bit for bit, not merely to several decimal places.
Does the identity still hold for angles past 90 degrees?
Yes. sin(2θ) = 2sinθcosθ holds for every real θ, including negative angles and angles beyond a full turn, because it descends from the addition formula, which places no restriction on either angle. At θ = 90°, both methods correctly return sin(180°) = 0.
What is the double-angle identity a special case of, beyond addition?
It is a first step in a family built by repeated addition: apply sin(x+y) again with one angle at 2θ and the other still at θ, and out comes triple-angle, then quadruple, and so on. Every multiple-angle identity for sine ultimately reduces to this same addition rule applied more than once.