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Why do trigonometric functions repeat their values?

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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.

Trigonometric functions repeat their values because they are periodic functions.12

When trigonometric functions are defined using a unit circle—a circle with a radius of 1 centered at the origin—the sine and cosine functions represent the coordinates of a point moving along that circle. As an angle increases or decreases, the terminal side of the angle eventually returns to the same position after completing a full rotation of 360° (or 2π radians).321

Because the point on the terminal side of the angle returns to the same coordinates after every full rotation, the ratios defining the trigonometric functions—such as the x-coordinate for cosine and the y-coordinate for sine—also return to their original values. Consequently, a trigonometric function has the same value for all coterminal angles, meaning the functions repeat their patterns at regular intervals. This inherent periodicity makes them essential for modeling periodic motion, such as sound waves, vibrations, and electromagnetic radiation.312