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How can sine 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 sine function defines the relationship between an interior angle and the ratio of two side lengths. Specifically, for an angle $A$ , the sine is defined as the ratio of the length of the side opposite that angle to the length of the hypotenuse:

$\sin A = \frac{opposite}{hypotenuse}$

To find a missing side, you can rearrange this formula based on the information you have:

To find the opposite side: If you know the angle $A$ and the hypotenuse $h$ , the length of the opposite side $a$ is calculated as:
$a = h \cdot \sin A$
To find the hypotenuse: If you know the angle $A$ and the opposite side $a$ , the length of the hypotenuse $h$ is calculated as:
$h = \frac{a}{\sin A}$

By using these algebraic rearrangements, you can solve for an unknown side length provided you have the measurement of one acute angle and one other side of the triangle.12