Question

Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos  (α - β)/2` is ______.

Options

• `(-6)/65`

• `3/sqrt(130)`

• `6/65`

• `-3/sqrt(130)`

Answer 1

Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos  (α - β)/2` is `underlinebb(-3/sqrt(130))`.

Explanation:

Given sin α + sin β = `-21/65` and cos α + cos β = `-27/65`

Squaring and adding ,we get

2(1 + cos α cos β + sin α sin β) = `1170/(65)^2`

2[1 + cos(α − β)] = `1170/(65)^2`

`cos^2  (α - β)/2 = 1170/(4 xx 65 xx 65) = (130 xx 9)/((130) xx (130))` 

`cos^2  (α - β)/2 = 9/130`

∴ `cos  (α - β)/2 = ±3/sqrt(130)`

Since, π < α – β < 3π

`π/2 < (α - β)/2 < (3π)/2`

So, `cos  ((α - β)/2) = -3/sqrt(130)`