sin2theta

Sin2θ, often written as sin(2θ), denotes the sine of the angle obtained by doubling θ. In trigonometry, it is a standard double-angle expression used to simplify and manipulate trigonometric expressions.

The most common form is the double-angle formula sin 2θ = 2 sin θ cos θ. This follows from the sine addition identity sin(α+β) = sin α cos β + cos α sin β by setting α and β equal to θ. Other useful forms include sin 2θ = 2 tan θ / (1 + tan^2 θ), valid when cos θ ≠ 0, obtained by dividing by cos^2 θ. The expression is sometimes related to cos 2θ through other identities, but sin 2θ itself is defined independently as 2 sin θ cos θ.

Key properties include its range and period. Since it is a sine function of 2θ, its values lie in [−1, 1], and its graph has period π with respect to θ. The zeros occur when 2θ = kπ, i.e., at θ = kπ/2. It attains a maximum of 1 when θ = π/4 + kπ, and a minimum of −1 when θ = 3π/4 + kπ.

Applications of sin 2θ appear across mathematics and physics. It is used to simplify products of sine and cosine, solve trigonometric equations, and appear in Fourier analysis and signal processing as the second harmonic. It also plays a role in geometry and rotations, where double-angle expressions describe relationships between angles and coordinates on the unit circle.