Question

If tan a = 1/2 evaluate 2sin a+cos a /
4cos a+3sin a​

Answer 1

Answer:

$\large\bold\red{\frac{4}{11}}$

Step-by-step explanation:

Given,

$\tan(a) = \frac{1}{2}$

Lets take a right angled triangle having,

perpendicular = p

base = b

hypotenus = h

Angle a is contained in the assumed triangle.

Therefore,

$\tan(a) = \frac{p}{b} = \frac{1}{2} \\ \\ = > b = 2p \: \: \: \: \: ..........(i)$

Also,

we know that,

in a triangle,

$h = \sqrt{ {p}^{2} + {b}^{2} } \\ \\ = > h = \sqrt{ {p}^{2} + {(2p)}^{2} } \\ \\ = > h = \sqrt{ {p}^{2} + 4 {p}^{2} } \\ \\ = > h = \sqrt{5 {p}^{2} } \\ \\ = > h = \sqrt{5} p$

Therefore,

$\sin(a) = \frac{p}{h} = \frac{p}{ \sqrt{5} p} = \frac{1}{ \sqrt{5} }$

and

$\cos(a) = \frac{b}{h} = \frac{2p}{ \sqrt{5} p} = \frac{2}{ \sqrt{5} }$

Therefore,

$\frac{2 \sin(a) + \cos(a) }{4 \cos(a) + 3 \sin(a) } = \frac{ \frac{2}{ \sqrt{5} } + \frac{2}{ \sqrt{5} } }{ \frac{8}{ \sqrt{5} } + \frac{3}{ \sqrt{5} } } \\ \\ = \frac{2 + 2}{8 + 3 } \\ \\ = \frac{4}{11}$

Hence,

$\bold{Value=\frac{4}{11}}$