Cosine
TrigonometryCosine is a trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle.
Formula
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
Definition
Cosine (written "cos") compares how wide a right triangle is to its longest side. It answers the question: "What fraction of the hypotenuse goes sideways?"
Example
With a $60^\circ$ angle and hypotenuse of $10$, $\cos(60^\circ) = 0.5$, so the adjacent side $= 0.5 \times 10 = 5$.
Key Insight
Sine measures "up" and cosine measures "across." Together they describe every point on a circle: $(\cos, \sin) = (\text{across}, \text{up})$.
Definition
For an angle $\theta$ in a right triangle, $\cos(\theta) = \text{adjacent}/\text{hypotenuse}$. On the unit circle, $\cos(\theta)$ equals the $x$-coordinate of the terminal point. Cosine is an even function: $\cos(-\theta) = \cos(\theta)$.
Example
$\cos(30^\circ) = \sqrt{3}/2 \approx 0.866$. For a ramp making a $30^\circ$ angle with the ground, the horizontal distance is $86.6\%$ of the ramp length.
Key Insight
Because cosine gives the $x$-coordinate on the unit circle, $\cos(0) = 1$ (fully to the right), $\cos(90^\circ) = 0$ (straight up), and $\cos(180^\circ) = -1$ (fully to the left). The pattern makes geometric sense.
Definition
$\cos(x) = \text{Re}(e^{ix}) = \sum (-1)^n x^{2n} / (2n)!$, convergent for all $x \in \mathbb{R}$. It is the unique solution to $f''(x) = -f(x)$ with $f(0) = 1$, $f'(0) = 0$. Cosine is an even function with period $2\pi$ and zeros at $\pi/2 + n\pi$ for integer $n$.
Example
$\cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + \ldots$ Using two terms, $\cos(0.1) \approx 1 - 0.005 = 0.995$, very close to the true value $0.99500417$.
Key Insight
The cosine rule in Euclidean geometry ($c^2 = a^2 + b^2 - 2ab\cos(C)$) is a direct generalization of the Pythagorean theorem and reduces to it when $C = 90^\circ$. Cosine also appears as the real part of complex multiplication.