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How do sine and cosine power series calculate values for angles?

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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 calculus, the sine and cosine functions can be expressed as infinite power series, which allow for the calculation of their values for any angle. To use these series, the angle must first be expressed in radians.123

The power series (specifically Maclaurin series) for these functions are:

Sine: $\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots$ 32
Cosine: $\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots$ 24

In these formulas, the exclamation point denotes the factorial function (e.g., $3! = 3 \times 2 \times 1 = 6$ ). These series converge for all real numbers $x$ , meaning that by calculating a sufficient number of terms, one can achieve any desired level of decimal precision for the sine or cosine of an angle. For example, substituting the radian equivalent of an angle into the series and summing the initial terms provides a highly accurate approximation of the function's value.321