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How are sine, cosine, and tangent calculated from the sides of a right triangle?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, the sine, cosine, and tangent functions are defined by the ratios of the side lengths of a right triangle relative to a given angle A.

Using the side lengths opposite to the angle (a), adjacent to the angle (b), and the hypotenuse (h), these functions are calculated as follows:

Sine (sin A): The ratio of the opposite side to the hypotenuse.
$\sin A = \frac{opposite}{hypotenuse} = \frac{a}{h}$
Cosine (cos A): The ratio of the adjacent side to the hypotenuse.
$\cos A = \frac{adjacent}{hypotenuse} = \frac{b}{h}$
Tangent (tan A): The ratio of the opposite side to the adjacent side.
$\tan A = \frac{opposite}{adjacent} = \frac{a}{b}$

These functions represent properties of the angle itself, remaining independent of the triangle's overall size.12