# How do you show that sin(x+60) + sin(x+120) =(√3)cosx ?

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Shreya Share
Feb 28, 2018

see below

#### Explanation:

We have,$\sin \left(x + 60\right) + \sin \left(x + 120\right)$ on L.H.S
According to formula, $\sin \left(a + b\right) = \sin a \cos b + \cos a \sin b$
Applying it,we get,
$\sin x \cos \left(60\right) + \cos x \sin \left(60\right) + \sin x \cos \left(120\right) + \cos x \sin \left(120\right)$
or, $\sin x / 2 + \cos x \times \left(\sqrt{3}\right) / 2 - \sin x / 2 + \cos x \times \left(\sqrt{3}\right) / 2$
{color(blue)(as cos "120 " is -1//2 and sin "120 " is (sqrt3)//2}

Thus,the expression comes out to be,$2 \times \left(\sqrt{3}\right) / 2 \times \cos x$ which is $\textcolor{red}{\sqrt{3} \cos x}$, the R.H.S

Then teach the underlying concepts
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Feb 28, 2018

#### Explanation:

$\sin \left(x + 60\right) + \sin \left(x + 120\right)$

$= 2 \cdot \sin \left\{\frac{\left(x + 60\right) + \left(x + 120\right)}{2}\right\} \cdot \cos \left\{\frac{\left(x + 60\right) - \left(x + 120\right)}{2}\right\}$

$= 2 \cdot \sin \left\{\frac{2 x + 180}{2}\right\} \cdot \cos \left\{\frac{- 60}{2}\right\}$

$= 2 \cdot \sin \left\{90 + x\right\} \cdot \cos \left\{30\right\}$

$= 2 \cdot \sin \left\{90 - \left(- x\right)\right\} \cdot \cos 30$

$= 2 \cdot \cos x \cdot \frac{\sqrt{3}}{2}$

$= \sqrt{3} \cdot \cos x$

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