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How can cosine be used to find a missing side 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 a right triangle, the cosine function is defined as the ratio of the length of the side adjacent to a given acute angle to the length of the hypotenuse.

To find a missing side using this relationship, you can use the formula:

$\cos (A) = \frac{adjacent}{hypotenuse}$

Depending on which side is unknown, you can rearrange the formula:

To find the adjacent side: Multiply the hypotenuse by the cosine of the angle:
$adjacent = hypotenuse \times \cos (A)$
To find the hypotenuse: Divide the adjacent side by the cosine of the angle:
$hypotenuse = \frac{adjacent}{\cos (A)}$

By knowing the measure of one acute angle and the length of one of these two sides, you can calculate the length of the other side using these trigonometric ratios.12