Question

tanA=b/a then find sin²A

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

${ \sin}^{2} \theta + { \cos}^{2} \theta = 1$

GIVEN

$\displaystyle \: \tan A = \frac{b}{a}$

TO DETERMINE

${ \sin}^{2}A$

EVALUATION

$\displaystyle \: \tan A = \frac{b}{a}$

$\implies \: \displaystyle \: \frac{ \sin A }{ \cos A\: } = \frac{b}{a}$

Let

$\sin A = bk \: \: and \: \cos A =ak$

Now

${ \sin }^{2} A + { \cos }^{2} A = 1$

$\implies \: {b}^{2} {k}^{2} + {a}^{2} { k}^{2} = 1$

$\implies \: {k}^{2}({b}^{2} + {a}^{2} )= 1$

$\displaystyle \: \implies \: {k}^{2}= \frac{1}{({b}^{2} + {a}^{2} )}$

RESULT

${ \sin}^{2}A$

$= \displaystyle \: {b}^{2} {k}^{2}$

$\displaystyle \: = \frac{ {b}^{2} }{({b}^{2} + {a}^{2} )}$