Why $\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ ?

Let $\sin x = a + bx + cx^2 + dx^3 + ex^4 + \cdots$ When $x = 0$ , $a=0$ Derivation of both sides at the same time: $\cos x = b + 2cx + 3dx^2 + 4dx^3 + \cdots$ When $x=0$ , then $b = 1 = \frac{1}{1!}$ Derivation of both sides at the same time again: $-\sin x = 2c + 6dx + 12dx^2 + \cdots$ When $x=0$ , then $c = 0$ Again: $-\cos x = 6d + 24ex + 120fx^2 + \cdots$ When $x=0$ , then $d = -\frac{1}{6} = -\frac{1}{3!}$ The same, $e = 0 , f = \frac{1}{5!} , g = 0 , h = -\frac{1}{7!} \cdots$ So $\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ Proven!

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

nice

시개경경목문 점클샵아 - 4 months, 2 weeks ago

:-)

Raymond Fang - 4 months, 2 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Why sin⁡x=x1!−x33!+x55!−x77!+⋯\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdotssinx=1!x​−3!x3​+5!x5​−7!x7​+⋯?

Comments

Why $\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ ?