# Power Reduction Formulas/Sine to 5th

(Redirected from Fifth Power of Sine)

## Theorem

$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$

where $\sin$ denotes sine.

## Proof

 $\displaystyle \sin 5 x$ $=$ $\displaystyle 5 \sin x - 20 \sin^3 x + 16 \sin^5 x$ Quintuple Angle Formula for Sine $\displaystyle \leadsto \ \$ $\displaystyle 16 \sin^5 x$ $=$ $\displaystyle \sin 5 x + 20 \sin^3 x - 5 \sin x$ rearranging $\displaystyle$ $=$ $\displaystyle \sin 5 x + 20 \paren {\frac {3 \sin x - \sin 3 x} 4} - 5 \sin x$ Power Reduction Formula for Cube of Sine $\displaystyle$ $=$ $\displaystyle \sin 5 x + 15 \sin x - 5 \sin 3 x - 5 \sin x$ multipying out $\displaystyle$ $=$ $\displaystyle 10 \sin x - 5 \sin 3 x + \sin 5 x$ rearranging $\displaystyle \leadsto \ \$ $\displaystyle \sin^5 x$ $=$ $\displaystyle \frac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$ dividing both sides by 16

$\blacksquare$