Question

If if sin theta equal to 4/5 find the value of sin 2 theta

Answer 1

$\large\underline\blue{\bold {Given:-  }}$

$\bullet \: \rm \: sin \: \theta \: = \dfrac{4}{5}$

$\large\underline\blue{\bold{To\:Find:-  }}$

$\bull \: \rm \: sin2 \theta \:$

$\begin{gathered}\Large{\bold{\green{\underline{Formula \: Used \::}}}} \end{gathered}$

$\rm :\implies\: \boxed{ \pink{ \bf \: cos \theta \: \: = \tt \: \sqrt{1 - {sin}^{2} } \theta \: }}$

$\rm :\implies\: \boxed{ \pink{ \bf \: sin2 \theta \: \: = \tt \: 2sin \theta \:cos \theta \:}}$

$\large\underline\purple{\bold{Solution :- }}$

Given that

$\rm :\implies\: \blue{sin \theta \: = \dfrac{4}{5} }$

So,

$\rm :\implies\:cos \theta \: = \sqrt{1 - {sin}^{2} \theta \:}$

$\rm :\implies\:cos \theta \: = \sqrt{1 - {(\dfrac{4}{5} )}^{2} }$

$\rm :\implies\:cos \theta \: = \sqrt{1 -\dfrac{16}{25} }$

$\rm :\implies\:cos \theta \: = \sqrt{\dfrac{25 - 16}{25} }$

$\rm :\implies\:cos \theta \: = \sqrt{\dfrac{9}{25} }$

$\rm :\implies\: \green{cos \theta \: = \dfrac{3}{5} }$

Now,

$\rm :\implies\:sin2 \theta \: = 2sin \theta \:cos \theta \:$

$\rm :\implies\:sin2 \theta \: = 2 \times \dfrac{4}{5} \times \dfrac{3}{5}$

$\rm :\implies\:sin2 \theta \: = \dfrac{24}{25}$