Sine
Sine is a trigonometric function defined as the ratio of the opposite side to the hypotenuse in a right triangle.
Formula
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
Definition
Sine (written "sin") compares how tall a right triangle is to the length of its hypotenuse, and is used to find a missing side or angle. For an acute angle $\theta$ in a right triangle, $\sin(\theta) = \text{opposite}/\text{hypotenuse}$; extended to the unit circle, $\sin(\theta)$ is the $y$-coordinate of the point on the circle at angle $\theta$ measured from the positive $x$-axis. Analytically, $\sin(x)$ is defined by the power series $\sum (-1)^n x^{2n+1} / (2n+1)!$, convergent for all $x \in \mathbb{R}$, and equivalently as $\sin(x) = \text{Im}(e^{ix})$ via Euler's formula; it is the unique solution to $f''(x) = -f(x)$ with $f(0) = 0$, $f'(0) = 1$.
Example
In a right triangle with a $30^\circ$ angle and hypotenuse $10$, $\sin(30^\circ) = 0.5$, so the opposite side $= 0.5 \times 10 = 5$; similarly, $\sin(45^\circ) = \sqrt{2}/2 \approx 0.707$, meaning a $45^\circ$ triangle's opposite leg is about $70.7\%$ of the hypotenuse. The series $\sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + \ldots$ converges for all real $x$ and is how calculators evaluate sine to arbitrary precision.
Key Insight
Sine is always between $-1$ and $1$, literally the height of a point on the unit circle above the center, which is why $\sin(0) = 0$, $\sin(90^\circ) = 1$, $\sin(180^\circ) = 0$, and $\sin(270^\circ) = -1$. As the imaginary part of the complex exponential, sine connects real analysis, complex analysis, and Fourier theory: every $L^2$ periodic function decomposes uniquely into sine and cosine components.