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  4. The value of 4sinα sin(α+(π/3))sin(α +(2π/3))=

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Q. The value of $4sin\alpha\,sin\left(\alpha+\frac{\pi}{3}\right)sin\left(\alpha +\frac{2\pi}{3}\right)=$

Trigonometric Functions

Solution:

We have, $4sin\alpha\,sin\left(\alpha+\frac{\pi}{3}\right)sin\left(\alpha +\frac{2\pi}{3}\right)$

$=2sin\alpha\left\{2\,sin\left(\alpha+\frac{2\pi}{3}\right)sin\left(\alpha +\frac{\pi }{3}\right)\right\}$

$=2sin\alpha\left[2sin\left(\alpha+120^{\circ}\right)sin\left(\alpha+60^{\circ}\right)\right]$
$=2sin\,\alpha [cos\left(\alpha+120^{\circ}-\alpha-60^{\circ}\right)-$
$cos(\alpha + 120^{\circ} + \alpha + 60^{\circ})]$
$= 2sina[cos60^{\circ} - cos(180^{\circ} + 2\alpha)]$
$=2sin\alpha\cdot\frac{1}{2}-2sin\alpha\left(-cos2\alpha\right)$
$= sin\alpha + 2cos2\alpha sin\alpha = sin\alpha + sin\left(2\alpha + \alpha\right) -$
$sin\left(2\alpha-\alpha\right)$
$= sin\alpha + sin3\alpha - sin\alpha = sin3\alpha$