Question

if 3tantheta =4, find the value of 4costheta - sin theta /2costheta + sin theta

Answer 1

$\bf \LARGE \it Hey \: User!!!$

given :-

3tan∅ = 4

therefore tan∅ = 4/3

consider any right angle triangle abc.

we know that tan∅ = perpendicular/base

▶BC = perpendicular = 4
▶AC = base = 3
▶AB = hypotenuse = ?

by Pythagoras theorem we get :-

>> AB = √(BC² + AC²)
>> AB = √(4² + 3²)
>> AB = √(16 + 9)
>> AB = √25
>> AB = 5

hence, the hypotenuse is 5.

now,

we know that sin∅ = perpendicular/hypotenuse and cos∅ = base/hypotenuse

▶sin∅ = 4/5
▶cos∅ = 3/5

$\rm = > \frac{4 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta} \\ \\ \rm = > \frac{(4 \times \frac{3}{5} ) - \frac{4}{5} }{(2 \times \frac{3}{5}) + \frac{4}{5} } \\ \\ \rm \large = > \frac{ \frac{12}{5} - \frac{4}{5} }{ \frac{6}{5} + \frac{4}{5} } \\ \\ \rm \large= > \frac{ \frac{8}{5} }{ \frac{10}{5} } \\ \\ \rm = > \frac{ \cancel8 }{ \cancel5} \times \frac{ \cancel5 }{ \cancel10 } \\ \\ \rm = > \boxed{ \frac{4}{5} } \: final \: answer : )$



$\bf \LARGE \it Cheers!!!$