In a right-angled triangle, the trigonometric functions relate the angles to the ratios of the triangle's sides. These functions — sine, cosine, tangent, and their reciprocals — are foundational in geometry and trigonometry.
\[ \sin \alpha = \frac{a}{c} = \frac{\text{opposite}}{\text{hypotenuse}} \]
The sine of angle \( \alpha \) is the ratio of the length of the side opposite to \( \alpha \) to the hypotenuse.
\[ \cos \alpha = \frac{b}{c} = \frac{\text{adjacent}}{\text{hypotenuse}} \]
The cosine of angle \( \alpha \) is the ratio of the length of the adjacent side to the hypotenuse.
\[ \tan \alpha = \frac{a}{b} = \frac{\text{opposite}}{\text{adjacent}} \]
The tangent of angle \( \alpha \) is the ratio of the opposite side to the adjacent side.
\[ \cot \alpha = \frac{b}{a} = \frac{\text{adjacent}}{\text{opposite}} \]
The cotangent of angle \( \alpha \) is the reciprocal of the tangent function.
\[ \text{SOH-CAH-TOA} \]
This mnemonic helps remember the trigonometric ratios:
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent