Trigonometric Ratio
A trigonometric ratio is a ratio of two sides of a right triangle that corresponds to a specific trigonometric function.
Definition
A trigonometric ratio is a ratio of two side lengths of a right triangle associated with a given angle. The six ratios are $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, and $\cot$, and because similar triangles have proportional sides, these ratios depend only on the angle, not the triangle's size, a consequence of the AA similarity theorem: all right triangles sharing an acute angle are similar, so their side ratios are equal. This invariance allows the functions $\sin, \cos, \tan: \mathbb{R} \to \mathbb{R}$ to be extended from acute angles to all reals via the unit circle or Taylor series.
Example
If a right triangle has an opposite side of $3$ and a hypotenuse of $5$, the sine ratio for that angle is $3/5 = 0.6$; two right triangles that both have a $50^\circ$ angle, one with sides $6$-$7.7$-$10$ and the other $3$-$3.86$-$5$, give the same ratio $\text{opposite}/\text{hypotenuse} = 0.6$ in both, confirming $\sin(50^\circ) = 0.766$. The identity $\sin(\theta)/\cos(\theta) = \tan(\theta)$ is both a geometric ratio and an algebraic identity between two analytic functions, illustrating how geometry and analysis are unified.
Key Insight
All triangles with the same angle have the same trig ratios, even at wildly different sizes, the ratio is like a fingerprint for each angle. Trig ratios generalize to hyperbolic ratios $\sinh(t)/\cosh(t) = \tanh(t)$ for the unit hyperbola $x^2 - y^2 = 1$, giving hyperbolic functions the same structural role in Minkowski geometry that trig ratios play in Euclidean geometry.