Question

Which of the following expressions is equivalent to sin 6x

Answer 1

$\textbf{Given:}$

$\sin6x$

$\textbf{To find:}$

$\text{Expression equivalent to \sin6x }$

$\textbf{Solution:}$

$\text{Consider,}$

$\sin6x$

$=\sin3(2x)$

$\text{Using the identity,}$

$\boxed{\bf\sin3A=3\,\sin\,A-4\,\sin^3A}$

$=3\,\sin2x-4\,\sin^32x$

$=3(2\,\sin{x}\cos{x})-4\,(2\,\sin{x}\,\cos{x})^3$

$=6\,\sin{x}\,\cos{x}-4\,(8\,\sin^3x\,\cos^3x)$

$=6\,\sin{x}\,\cos{x}-32\,\sin^3x\,\cos^3x$

$\textbf{Answer:}$

$\boxed{\bf\,\sin6x=6\,\sin{x}\,\cos{x}-32\,\sin^3x\,\cos^3x}$