Sine
TrigonometrySine 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") is a math tool that compares how tall a right triangle is to how long its longest side (hypotenuse) is. You use it to find a missing side or angle in a right triangle.
Example
In a right triangle with a $30^\circ$ angle and a hypotenuse of $10$, $\sin(30^\circ) = 0.5$, so the opposite side $= 0.5 \times 10 = 5$.
Key Insight
Sine is always between $-1$ and $1$. Think of it as a fraction of how high a point is on a circle compared to the circle's radius.
Definition
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 unit circle at angle $\theta$ measured from the positive $x$-axis.
Example
$\sin(45^\circ) = \sqrt{2}/2 \approx 0.707$. This means a $45^\circ$ right triangle's opposite leg is about $70.7\%$ of the hypotenuse length.
Key Insight
Because the unit circle has radius $1$, the sine value literally equals the height of the corresponding point on the circle. This is why $\sin(0) = 0$, $\sin(90^\circ) = 1$, $\sin(180^\circ) = 0$, $\sin(270^\circ) = -1$.
Definition
Analytically, $\sin(x)$ is defined by the power series $\sum (-1)^n x^{2n+1} / (2n+1)!$, convergent for all $x \in \mathbb{R}$. Equivalently, $\sin(x) = \text{Im}(e^{ix})$ via Euler's formula. It is the unique solution to $f''(x) = -f(x)$ with $f(0) = 0$, $f'(0) = 1$.
Example
$\sin(x) = x - x^3/6 + x^5/120 - x^7/5040 + \ldots$ This series converges for all real $x$ and is used in calculators to evaluate sine to arbitrary precision.
Key Insight
The sine function is the imaginary part of the complex exponential, connecting real analysis, complex analysis, and Fourier theory. Every $L^2$ periodic function decomposes uniquely into sine and cosine components.