Question

If 5 tan θ = 3, then what is the value of `((5 sin θ - 3 cos θ)/(4 sin θ + 3 cos θ))`?

Answer 1

Given, 5 tan θ = 3

∴ `tan θ = 3/5`

AC² = AB² + BC²

= (3k)² + 25k²

∴ `AC = sqrt(34)k`

So, `cos θ = 5/sqrt(34)`

And `sin θ = 3/sqrt(34)`

∴ `(5 sin θ - 3 cos θ)/(4 sin θ + 3 cos θ)`

= `(5(3/sqrt(34)) - 3(5/sqrt(34)))/(4(3/sqrt(34)) + 3(5/sqrt(34))`

= `((15 - 15)/(sqrt(34)))/((12 + 15)/sqrt(34))`

= `(0/sqrt(34))/(27/sqrt(34))`

= 0

Answer 2

Given, 5 tan θ = 3

∴ `tan θ = 3/5`

`(5 sin θ - 3 cos θ)/(4 sin θ + 3 cos θ)`

Divide numerator and denominator by cos θ

⇒ `((5 sin θ)/cos θ - (3 cos θ)/cos θ)/((4 sin θ)/cos θ + (3 cos θ)/ cos θ)`

⇒ `(5 tan θ - 3)/(4 tan θ + 3)`

⇒ `(5 xx 3/5 - 3)/(4 xx 3/5 + 3)`

= `(3 - 3)/(12/5 + 3)`

= `0/(27/5)`

= 0