Power Series Expansion for Sine Function

Theorem

The sine function has the power series expansion:

$\ds \sin x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$

$\ds $

$=$

$\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots$

valid for all $x \in \R$.

Proof

From Derivative of Sine Function:

$\dfrac \d {\d x} \sin x = \cos x$

From Derivative of Cosine Function:

$\dfrac \d {\d x} \cos x = -\sin x$

Hence:

$\ds \dfrac {\d^2} {\d x^2} \sin x$

$=$

$\ds -\sin x$

$\ds \dfrac {\d^3} {\d x^3} \sin x$

$=$

$\ds -\cos x$

$\ds \dfrac {\d^4} {\d x^4} \sin x$

$=$

$\ds \sin x$

and so for all $m \in \N$:

$\ds m = 4 k: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sin x$

$=$

$\ds \sin x$

$\ds m = 4 k + 1: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sin x$

$=$

$\ds \cos x$

$\ds m = 4 k + 2: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sin x$

$=$

$\ds -\sin x$

$\ds m = 4 k + 3: \ \ $

$\ds \dfrac {\d^m} {\d x^m} \sin x$

$=$

$\ds -\cos x$

where $k \in \Z$.

This leads to the Maclaurin series expansion:

$\ds \sin x$

$=$

$\ds \sum_{r \mathop = 0}^\infty \paren {\frac {x^{4 k} } {\paren {4 k}!} \map \sin 0 + \frac {x^{4 k + 1} } {\paren {4 k + 1}!} \map \cos 0 - \frac {x^{4 k + 2} } {\paren {4 k + 2}!} \map \sin 0 - \frac {x^{4 k + 3} } {\paren {4 k + 3}!} \map \cos 0}$

$\ds $

$=$

$\ds \sum_{r \mathop = 0}^\infty \paren {\frac {x^{4 k + 1} } {\paren {4 k + 1}!} - \frac {x^{4 k + 3} } {\paren {4 k + 3}!} }$

Sine of Zero is Zero, Cosine of Zero is One

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$

setting $n = 2 k$

From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.

$\blacksquare$

Also see

Power Series Expansion for Cosine Function

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Trigonometric Functions: $20.21$
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3.2$: Power series: $(1.46)$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): power series
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): power series
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.21.$