Question

If cos θ : sin θ = 1 : 2, then find the value of `(8costheta - 2sintheta)/(4costheta + 2sintheta`

Answer 1

cos θ : sin θ = 1 : 2

The value of `(8costheta - 2sintheta)/(4costheta + 2sintheta`

= `(8(sintheta/2) - 2sintheta)/(4(sintheta/2) + 2sintheta)`

= `(4 sintheta - 2 sintheta)/(2 sintheta + 2 sintheta)`

= `(2 sintheta)/(4 sintheta)`

= `2/4`

= `1/2`

Aliter:

cos θ : sin θ = 1 : 2

 cos θ = sin θ

⇒ 2 = `sintheta/costheta`

⇒ 2 = tan θ

AC = `sqrt(1^2 + 2^2)`

= `sqrt(1 + 4)`

⇒ `sqrt(5)`

∴ sin θ = `2/sqrt(5)`, cos θ = `1/sqrt(5)`

The value of `(8 cos theta - 2 sin theta)/(4 cos theta + 2 sin theta)`

= `8(1/sqrt(5)) - 2(2/sqrt(5)) ÷ 4(1/sqrt(5)) + 2 xx (2/sqrt(5))`

= `(8/sqrt(5)) - (4/sqrt(5)) ÷ (4/sqrt(5)) + (4/sqrt(5))`

= `((8 - 4)/sqrt(5)) ÷ ((4 + 4)/sqrt(5))`

= `(4/sqrt(5)) ÷ (8/sqrt(5))`

= `(4/sqrt(5)) xx (sqrt(5)/8)`

= `(4/8)`

= `1/2`

∴ The value is `1/2`