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Why are radians useful in mathematics and physics?

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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 radian is considered the most convenient unit for theoretical work because it simplifies the relationship between angles and circular geometry.1

A radian is defined as the angle at the center of a circle that intercepts an arc equal in length to the circle's radius. Because of this definition, the length of an arc is directly proportional to the radius and the angle in radians, allowing for cleaner mathematical expressions without the conversion factors required by degrees.213

In mathematics and physics, using radians is advantageous because it makes the derivatives and integrals of trigonometric functions more straightforward. For instance, the derivative of $\sin (x)$ is simply $\cos (x)$ only when $x$ is measured in radians. If degrees were used, these formulas would require additional constants involving $π$ and 180. By defining a complete revolution as $2 π$ radians, the unit aligns naturally with the properties of circles and periodic functions.1