Question

If 3 cot θ = 4, find the value of \[\frac{4 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta}\]

Answer 1

We have: 

`3 cot θ=4` 

`cotθ= 4/3` 

Since we know that in right angle triangle 

`cot θ=" Base"/"Perpendicular"` 

`cot θ=" Base"/ "Hypotenuse"` 

`sinθ = "Prependicular"/ "Hypotenuse" `  

`"Hypotenuse"= sqrt(("Perpendicular")^2+("Base")^2)` 

`"Hypotenuse"=sqrt((3)^2+(4)^2)` 

`"Hypotenuse"=sqrt25` 

`"Hypotenuse"=5` 

Now, we find `(4 cosθ- sin θ)/(2 cos θ+sin θ)` 

⇒ `(4 cosθ- sin θ)/(2 cos θ+sin θ)=(4xx 4/5-3/5)/(2xx4/5+3/5)` 

⇒`(4 cosθ- sin θ)/(2 cos θ+sin θ) (16/5-3/5)/(8/5+3/5)`  

⇒`(4 cosθ- sin θ)/(2 cos θ+sin θ)=``(13/5)/(11/5)` 

⇒`(4 cosθ- sin θ)/(2 cos θ+sin θ) = 13/11` 

Hence the value of  `(4 cosθ- sinθ)/(2 cos θ+sin θ) "is" 13/11`