Question

Find an acute angle θ when `(cos θ - sin θ)/(cos θ + sin θ) = (1 - sqrt(3))/(1 + sqrt(3))`

Answer 1

Given, `(cos θ - sin θ)/(cos θ + sin θ) = (1 - sqrt(3))/(1 + sqrt(3))`

Cross multiplying

`(1 + sqrt(3)) (cos θ - sin θ) = (1 - sqrt(3)) (cos θ + sin θ)`

`1(cos θ - sin θ) + sqrt(3)(cos θ - sin θ) = 1(cosθ + sin θ) - sqrt(3) (cos θ + sin θ)`

`cos θ - sin θ + sqrt(3)cos θ - sqrt(3)sin θ = cos θ + sin θ - sqrt(3) cos θ - sqrt(3)sin θ - sin θ + sqrt(3)cos θ = sin θ - sqrt(3)cos θ`

`sqrt(3)cos θ + sqrt(3) cos θ` = sin θ + sin θ

`2sqrt(3)cos θ` = 2 sin θ

`sqrt(3)cos θ` = sin θ

`sqrt(3) = sinθ/cosθ`

tan θ = `sqrt(3)`

Since tan 60° = `sqrt(3)`

Therefore, θ = 60°