Question

sin 20° - sin 100° + sin 140°
​

Answer 1

Answer:

Sin (20-100-140)=sin 60

Answer 2

$\large\underline{\sf{Solution-}}$

Given Trigonometric expression is

$\rm :\longmapsto\:sin20\degree - sin100\degree + sin140\degree$

can be re-arranged as

$\rm \:  =  \:( sin140\degree + sin20\degree ) - sin100\degree$

We know,

$\purple{\boxed{\tt{ sinx + siny = 2sin\bigg[\dfrac{x + y}{2} \bigg]cos\bigg[\dfrac{x - y}{2} \bigg]}}}$

So, using this, we get

$\rm \:  =  \: 2sin\bigg( \dfrac{140\degree + 20\degree }{2} \bigg)cos\bigg( \dfrac{140\degree - 20\degree }{2} \bigg) - sin100\degree$

$\rm \:  =  \: 2sin\bigg( \dfrac{160\degree }{2} \bigg)cos\bigg( \dfrac{120\degree }{2} \bigg) - sin100\degree$

$\rm \:  =  \: 2sin80\degree cos60\degree - sin100\degree$

$\rm \:  =  \: 2sin80\degree \times \dfrac{1}{2} - sin100\degree$

$\rm \:  =  \: sin80\degree - sin100\degree$

$\rm \:  =  \: sin(180\degree - 100\degree ) - sin100\degree$

We know,

$\purple{\boxed{\tt{ \: \: sin(180\degree - x) \: = \: sinx \: \: }}}$

So, using this, we get

$\rm \:  =  \: sin100\degree - sin100\degree$

$\rm \:  =  \: 0$

Hence,

$\purple{\boxed{\tt{ \: \: \: sin20\degree - sin100\degree + sin140\degree = 0 \: \: \: }}}$

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LEARN MORE

$\purple{\boxed{\tt{ sinx - siny = 2cos\bigg( \dfrac{x + y}{2} \bigg)sin\bigg( \dfrac{x - y}{2} \bigg)}}}$

$\purple{\boxed{\tt{ cosx + cosy = 2cos\bigg( \dfrac{x + y}{2} \bigg)cos\bigg( \dfrac{x - y}{2} \bigg)}}}$

$\purple{\boxed{\tt{ cosx - cosy = - 2sin\bigg( \dfrac{x + y}{2} \bigg)sin\bigg( \dfrac{x - y}{2} \bigg)}}}$