Question

If 5 tan θ = 4, find the value of `(5 sin θ − 3 cos θ)/(5 sin θ + 2 cos θ)`

Answer 1

Given,

Opposite side = 4, Adjacent side = 5

Hypotenuse = `sqrt((4)^2 + (5)^2)`

= `sqrt (16^2 + 25^2)`

Hypotenuse = `sqrt41`

sin θ = `"opposite"/"hypotenuse" = 4/sqrt41, cos θ = "adjacent"/"hypotenuse" = 5/sqrt41`

= `(5 sin θ − 3 cos θ)/(5 sin θ + 2 cos θ)`

= `(5· 4/sqrt41 − 3· 5/sqrt 41)/(5· 4/sqrt41 + 2· 5/sqrt 41)`

Factor ​ `1/ sqrt41` from the numerator and denominator (it cancels):

= `(20 − 15)/(20+ 10)`

= `5/30`

= `1/6`