Sine ($\sin$)

For a real angle $\theta$, $\sin \theta$ is the vertical coordinate on the unit circle reached by a counterclockwise rotation through $\theta$ radians. For an acute angle in a right triangle, it is the opposite side divided by the hypotenuse. In particular,The first-quarter rise and second-quarter fall are visible in the circle projection; the lower semicircle gives negative values.

Reflecting the unit circle across the horizontal axis gives $\sin (-\theta )=-\sin \theta$, so sine is an odd function. A full turn gives $\sin (\theta +2\pi )=\sin \theta$, making it a periodic function. For real arguments, its range is $[-1,1]$ and its zeros of a function occur at $\theta =k\pi$ for integers $k$. The cosine describes the perpendicular coordinate of the same point, and the Pythagorean trigonometric identity keeps their two values tied together.

The sine function extends to an entire function on the complex plane by the displayed exponential formula. Substituting $z=x+iy$ and grouping the real and imaginary terms gives $\sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$, for real $x,y$. Thus its values grow exponentially in imaginary directions, even though the real sine is bounded.

The paired zero factors converge locally uniformly because $\sum n^{-2}$ converges. Hadamard factorization for the order-one entire sine function gives a possible exponential multiplier; oddness removes its linear exponent and the derivative at zero fixes its constant. This product turns a Dirichlet oscillator determinant ratio into an elementary sine ratio.