Hypotenuse
The hypotenuse is the longest side of a right triangle, located opposite the right angle.
Formula
c = \sqrt{a^2 + b^2}
Definition
The hypotenuse is the longest side of a right triangle, the one directly across from the right angle (the corner with the little square). Its length $c$ is found from the two legs $a$ and $b$ using the Pythagorean theorem, $c = \sqrt{a^2 + b^2}$, a relationship that follows from the Euclidean metric. On the unit circle, the reference right triangle's hypotenuse is always $1$, which makes $\sin(t)$ and $\cos(t)$ the legs directly, and explains why $\sin^2(t) + \cos^2(t) = 1$.
Example
In a triangle with sides $3$, $4$, and $5$, the side of length $5$ is the hypotenuse because it sits across from the $90$-degree corner; likewise, a triangle with legs $5$ and $12$ has hypotenuse $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$. In the complex plane, the modulus $|z| = \sqrt{a^2 + b^2}$ for $z = a + bi$ is precisely the hypotenuse of the right triangle formed by the real and imaginary parts, a relationship that generalizes to inner product spaces via $||v||^2 = \langle v, v \rangle$.
Key Insight
No matter how you flip or rotate a right triangle, the hypotenuse is always the side opposite the right angle, and it serves as the denominator in both the sine and cosine ratios: $\sin = \text{opposite}/\text{hypotenuse}$ and $\cos = \text{adjacent}/\text{hypotenuse}$. It also connects to the geometric mean altitude theorem, relating the altitude and the segments it creates on the hypotenuse, a result with deep implications in projective geometry.