Power Reduction Formulas/Sine to 5th
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Theorem
- $\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$
where $\sin$ denotes sine.
Proof
\(\ds \sin 5 x\) | \(=\) | \(\ds 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\) | Quintuple Angle Formula for Sine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 16 \sin^5 x\) | \(=\) | \(\ds \sin 5 x + 20 \sin^3 x - 5 \sin x\) | rearranging | ||||||||||
\(\ds \) | \(=\) | \(\ds \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} - 5 \sin x\) | Power Reduction Formula for Cube of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin 5 x + 15 \sin x - 5 \sin 3 x - 5 \sin x\) | multipying out | |||||||||||
\(\ds \) | \(=\) | \(\ds 10 \sin x - 5 \sin 3 x + \sin 5 x\) | rearranging | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin^5 x\) | \(=\) | \(\ds \frac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}\) | dividing both sides by 16 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.59$