Question

if tantheta =4/3,find the value of 2sintheta+2costheta/3sintheta-2costheta​

Answer 1

2sintheta +2costheta = 2

3sintheta - 2costheta = -2

=2/-2

= -1

Answer 2

we know that,

$\tan(theta) = \frac{4}{3} = \frac{side \: opp \: .to \: theta}{side \: adj. \: to \: thteta}$

see the given figure,

now by Pythagoras theorem,

hypotenuse=

$\sqrt{4 {}^{2} + {3}^{2} } \\ \sqrt{16 + 9 } \\ \sqrt{25} \\ 5 \: units$

now,

$\sin(theta) = \frac{side \: opp. \: to \: theta \: }{hypotenuse} = \frac{4}{5}$

and

$\cos(theta) = \frac{side \: adj. \: to \: theta \: }{hypotenuse} = \frac{3}{5}$

so,

$\frac{2 \sin(theta) + 2 \cos(theta) }{3 \sin(theta) - 2 \cos(theta) }$

$\frac{2 \times ( \frac{4}{5} ) + 2 \times ( \frac{3}{5}) }{3 \times ( \frac{4}{5}) - 2 \times (\frac{3}{5} )}$

$= \frac{ \frac{8}{5} + \frac{6}{5} }{ \frac{12}{5} - \frac{6}{5} }$

$= \frac{ \frac{8 + 6}{5} }{ \frac{12 - 6}{5} }$

$= \frac{ \frac{14}{5} }{ \frac{6}{5} }$

$= \frac{14}{5} \times \frac{5}{6}$

$\frac{14}{6}$

$\frac{7}{3}$

so,

$\frac{7}{3}$