Introduction to Trigonometry - Trigonometric Ratios

Trigonometric ratios are defined using a right-angled triangle. Imagine a triangle $ABC$ with a right angle at $B$. For an acute angle $\theta$ at vertex $A$, the side $BC$ is the 'Opposite' side (Perpendicular), $AB$ is the 'Adjacent' side (Base), and $AC$ is the 'Hypotenuse' (the longest side, opposite the $90^\circ$ angle).

There are six fundamental trigonometric ratios that relate the sides of a right triangle to its angles: sine ($\sin$), cosine ($\cos$), tangent ($\tan$), cosecant ($\text{cosec}$), secant ($\sec$), and cotangent ($\cot$).

The 'Opposite' and 'Adjacent' sides are relative to the angle being considered. If you switch focus from $\angle A$ to $\angle C$ in $\triangle ABC$, the side $AB$ (previously the Base for $\angle A$) becomes the Perpendicular for $\angle C$, while $BC$ becomes the Base.

The reciprocal relationships are crucial: $\text{cosec } \theta$ is the reciprocal of $\sin \theta$, $\sec \theta$ is the reciprocal of $\cos \theta$, and $\cot \theta$ is the reciprocal of $\tan \theta$.

The values of $\sin \theta$ and $\cos \theta$ are always between $0$ and $1$ (for acute angles) because they are ratios where the Hypotenuse (the longest side) is in the denominator. A ratio where the denominator is the largest side can never exceed $1$.

The tangent and cotangent ratios can also be expressed as quotients of sine and cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Standard angles for trigonometric ratios include $0^\circ, 30^\circ, 45^\circ, 60^\circ$, and $90^\circ$. Visualizing a table or the unit circle helps remember that $\sin$ increases from $0$ to $1$ as the angle increases, while $\cos$ decreases from $1$ to $0$.