How this instrument works
sin(2θ) = 2 sinθ cosθ says something specific: the sine of a doubled angle is not twice the original sine, but twice the product of that angle's sine and cosine. It falls out of the angle-sum formula sin(A+B) = sinA cosB + cosA sinB the moment both angles are set equal to θ, since sin(θ+θ) collapses to sinθcosθ + cosθsinθ, which is just 2 sinθ cosθ written once instead of twice.
There is a tidy picture behind the algebra. Take a triangle with two sides of length 1 meeting at an angle of 2θ; its area, by the standard half-ab-sinC rule, is ½ sin(2θ). Drop the altitude from that apex and the triangle splits into two identical right triangles, each with hypotenuse 1 and base angle θ — their combined area is sinθ times cosθ. Setting the two area expressions equal is the whole identity, with no algebra beyond a triangle cut in half.
The shape of the curve holds a surprise: sin(2θ) reaches its ceiling of 1 not when θ is largest but at θ = 45°, where sinθ and cosθ are equal and their product peaks. Push θ past 45° and sin(2θ) actually falls back toward zero by θ = 90°, even though θ itself kept climbing — doubling the angle does not mean doubling, or even growing, the sine.
- Enter your angle into the 'Angle, θ' field; the unit selector beside it defaults to degrees and also accepts radians or turns.
- The sheet doubles θ internally and takes its sine — you never have to compute 2θ or look up a second angle yourself.
- Read the result in the 'sin(2θ)' field, shown to eight decimal places and always a number between −1 and 1.
- Change the unit selector to radians or turns to see the same angle expressed differently, without retyping the value.
Worked example — a mirror rotated thirty degrees
A scanning mirror at rest is tilted by θ = 30° (0.5235987755982988 rad) to steer a laser beam. A flat mirror's reflection rule says rotating the mirror by θ swings the reflected beam through twice that angle, 2θ = 60°, so the beam's sideways shift depends on sin(60°) rather than on sin(30°) itself.
Entering θ = 0.5235987755982988 radians (30°) into the 'Angle, θ' field returns sin(2θ) = 0.8660254037844386 in the 'sin(2θ)' field, shown rounded to 0.86602540. That is exactly 2 × sin(30°) × cos(30°) = 2 × 0.5 × 0.8660254037844387, since sin(30°) equals one half and leaves the doubled product equal to cos(30°) itself.
Questions
What does sin(2θ) equal in terms of θ alone?
sin(2θ) = 2 sinθ cosθ. Rather than the sine and cosine of a whole new angle 2θ, the doubled angle's sine comes purely from θ's own sine and cosine — no separate lookup of 2θ needed, since doubling and multiplying trade places inside this identity.
Where does the double-angle formula come from?
It is the angle-sum formula sin(A+B) = sinA cosB + cosA sinB with A and B both set to θ. That turns sin(θ+θ) into sinθcosθ + cosθsinθ, and adding those two identical terms gives 2 sinθ cosθ — the doubling formula is addition's special case when both angles match.
Is sin(2θ) the same thing as 2 sinθ?
No, and mixing them up is the most common mistake with this identity. sin(2θ) equals 2 sinθ cosθ, with an extra cosθ factor attached. At θ = 30°, 2 sin(30°) would be 1, but sin(60°) is only about 0.866 — the cosine term is doing real work, not just along for the ride.
Why does sin(2θ) peak at θ = 45° instead of at a larger angle?
Because sin(2θ) equals 1 exactly when 2θ = 90°, and that happens at θ = 45° — the one point where sinθ and cosθ are equal, so their product is largest. Past 45°, cosθ shrinks faster than sinθ grows, so the product, and sin(2θ) with it, comes back down toward zero by θ = 90°.
Does the formula still work for angles outside 0° to 90°?
Yes — sin(2θ) = 2 sinθ cosθ holds for every real θ, including negative angles and angles past 360°, because sine and cosine are defined for all real inputs. The identity does not depend on θ sitting in any particular quadrant or staying within a single turn.
How does this relate to the general angle-addition formula for sin(A+B)?
It is that same formula with B replaced by θ, so A and B are equal. Angle addition is the general tool for combining any two angles; the double-angle version applies that tool to an angle and itself, which is why the cross terms sinθcosθ and cosθsinθ turn out identical and simply add.