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How are cosecant and sine related?

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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 cosecant (csc) and sine (sin) functions are reciprocals of each other.12

For any angle $θ$ where the sine is not zero, their relationship is defined as:

$\csc θ = \frac{1}{\sin θ}$

Conversely, this also means:

$\sin θ = \frac{1}{\csc θ}$

In the context of a right triangle, these functions represent ratios of the triangle's sides. While the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, the cosecant is the ratio of the length of the hypotenuse to the length of the side opposite the angle.13