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Why do similar right triangles have the same sine, cosine, and tangent values for matching angles?

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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.

Similar right triangles have the same sine, cosine, and tangent values because these trigonometric functions are defined by the ratios of side lengths, which remain constant in similar figures.12

In Euclidean geometry, two triangles are considered similar if they share the same shape, meaning their corresponding angles are equal. According to the AAA (angle-angle-angle) similarity theorem, when corresponding angles are equal, the corresponding sides of the triangles are proportional.13

Because trigonometric functions like sine, cosine, and tangent are defined as ratios of these sides (e.g., $\sin A = opposite / hypotenuse$ ), the scaling factor between similar triangles cancels out. For instance, if one triangle is a scaled version of another by a factor of $s$ , both the numerator and denominator of the ratio are multiplied by $s$ , leaving the overall value of the trigonometric function unchanged. Consequently, these values depend solely on the angle itself rather than the size of the triangle.123