How to Calculate sin(2θ) — The Double-Angle Formula
The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). It lets you find the sine of a doubled angle directly from the sine and cosine of the original angle, without evaluating sin(2θ) as a separate angle. This calculator takes your angle θ, computes sin(θ) and cos(θ), and multiplies them together (times 2) to get sin(2θ), while also reporting the doubled angle itself.
Deriving the identity
The double-angle formula falls directly out of the angle-addition formula for sine: sin(A + B) = sin A cos B + cos A sin B. Setting A = B = θ gives sin(θ + θ) = sin θ cos θ + cos θ sin θ, which simplifies to sin(2θ) = 2 sin(θ) cos(θ). Because it is built from the same sine and cosine values as θ, sin(2θ) is fully determined once you know θ — there is no ambiguity or second solution to pick between.
Degrees vs. radians
Choose the unit that matches your problem before entering the angle. Calculators and spreadsheets default to radians internally, so this tool converts degrees to radians (radians = degrees × π/180) before applying sin() and cos(), then converts the doubled angle back to whichever unit you selected. A full circle is 2π radians = 360°, so 2θ can exceed 360° or 2π for large starting angles — that is expected and does not affect the sin(2θ) result, since sine is periodic.
Common mistakes
- sin(2θ) ≠ 2 sin(θ): the cos(θ) factor cannot be dropped. At θ = 30°, 2 sin(θ) = 1 but sin(2θ) = sin(60°) ≈ 0.8660.
- Mixing degrees and radians: entering an angle in degrees while a formula expects radians (or vice versa) silently produces the wrong sine and cosine values.
- Assuming a wider range: sin(2θ) still only ranges from -1 to 1, even though 2θ itself can be a much larger angle than θ.