Question

Let α and 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 ______.

विकल्प

• `-3/sqrt(130)`

• `3/sqrt(130)`

• `6/65`

• `-6/65`

Answer 1

Let α and 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`  ...(i)

and cos α + cos β = `-27/65`  ...(ii)

On squaring and adding equations (i) and (ii), we get

(sin α + sin β)² + (cos α + cos β)² = `(-21/65)^2 + (-27/65)^2`

`\implies` sin²α + sin² β + 2 sin α sin β + cos² α + cos² β + 2 cos α cos β = `441/4225 + 729/4225`

`\implies` 2 + 2 cos (α – β) = `1170/4225`

`\implies` `2[2 cos^2 ((α - β)/2)] = 1170/4225`

`\implies cos^2 ((α - β)/2) = 1170/(4 xx 4225) = 9/130`

`\implies cos ((α - β)/2) = -3/sqrt(130)`  ...[∵ π < α – β < 3π]