How this instrument works
A double-angle formula is what falls out of the ordinary angle-addition identities the moment the two angles being added happen to be equal. sin(a+b) = sin a cos b + cos a sin b becomes sin(2θ) = 2 sinθ cosθ the instant b is set equal to a = θ; the same substitution in cos(a+b) = cos a cos b − sin a sin b produces cos(2θ) = cos²θ − sin²θ, and dividing the two results gives the tangent version. Nothing new is being invented here — doubling an angle is addition with itself, and these three lines are just that special case written out so nobody has to re-derive it each time.
This sheet bundles all three because a single θ decides all three outputs at once, yet a problem rarely needs only one of them. A projectile-range formula wants sin(2θ) alone, a power-reduction substitution wants cos(2θ), and a slope-doubling question wants tan(2θ) — feed the angle in once here and all three sit side by side instead of being looked up as three separate calculations. cos(2θ) is worth a second look on its own: because sin²θ + cos²θ = 1 always, cos²θ − sin²θ is secretly interchangeable with 2cos²θ − 1 and with 1 − 2sin²θ, three different-looking expressions for the identical number.
The tangent line hides a genuine trap. Its denominator, 1 − tan²θ, reaches zero whenever tanθ = ±1 — at 45°, 135°, and every 90° step beyond — and dividing by zero there is not a rounding glitch but a real statement that tan(2θ) is undefined, since 2θ then lands on 90° or 270°, where tangent itself has no finite value. This instrument's own check catches that denominator before it divides, rather than returning a silently wrong number.
- Enter your angle into the Angle, θ field.
- Pick degrees, radians, or turns from the unit selector next to it — degrees is the default.
- Read sin(2θ), cos(2θ), and tan(2θ) together; all three are computed from that same θ in one pass.
- Watch for the warning near 45°, 135°, and similar angles, where tan(2θ) is flagged as undefined rather than shown as a huge or wrong number.
Worked example — doubling a 30° angle
Set Angle, θ to 30 with the unit selector on degrees. Internally the sheet converts that to 0.5235987755982988 radians before evaluating anything, and the three lines return sin(2θ) = 0.86602540, cos(2θ) = 0.50000000, and tan(2θ) = 1.73205081 — precisely the sine, cosine, and tangent of 60°, because doubling 30° lands exactly there.
Each figure checks against the others without extra work. The point at 60° on the unit circle sits at (0.5, 0.8660254…), matching cos(2θ) and sin(2θ) exactly, and dividing the two, 0.8660254 ⁄ 0.5, returns 1.7320508 — the same tan(2θ) the third formula produced on its own, confirming the identity rather than merely trusting it.
Questions
How is the double-angle formula derived?
By setting both angles equal to θ inside the ordinary angle-addition identities. sin(θ+θ) expands into 2 sinθ cosθ, and cos(θ+θ) expands into cos²θ − sin²θ; the tangent line follows from dividing one result by the other. No separate proof is needed — doubling an angle is just adding it to itself.
Why does cos(2θ) also appear written as 2cos²θ − 1 or 1 − 2sin²θ?
Because sin²θ + cos²θ = 1 always holds, so sin²θ can be swapped for 1 − cos²θ or vice versa inside cos²θ − sin²θ. All three expressions return the identical number; this sheet computes the cos²θ − sin²θ form since it needs both sinθ and cosθ already, for the sin(2θ) line beside it.
Why does tan(2θ) sometimes come back undefined?
Its formula divides by 1 − tan²θ, which hits zero exactly where tanθ = ±1 — at 45°, 135°, 225°, and 315°. Doubling any of those angles lands on 90° or 270°, where tangent itself has no finite value, so the undefined result is mathematically correct, not a computation error. This sheet's check flags it before dividing.
What does this calculator offer beyond a plain sin(2θ) sheet?
One angle typed in returns sin(2θ), cos(2θ), and tan(2θ) together, rather than requiring the same θ re-entered three separate times across three separate tools. That matters whenever a problem needs more than one doubled value at once — a range equation paired with a slope check, say — instead of only ever needing sine on its own.
Does switching between degrees, radians, and turns change the results?
No — it only changes how θ is entered. The sheet converts to radians internally before evaluating any of the three formulas, so 30°, π⁄6 radians, and 1⁄12 turn all feed in the identical angle and produce the identical sin(2θ), cos(2θ), tan(2θ) triple.
Are these formulas special cases of a more general pattern?
Yes — they are the n = 2 instance of the multiple-angle formulas, which extend to sin(3θ), cos(4θ), and beyond using repeated angle addition or de Moivre's theorem. The double-angle case is simply the first and most-used step of that ladder.