Question

Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2`  and sin y = `- 24/25` with `pi < y < (3pi)/2`

Answer 1

cos x = `- 4/5`

`pi < x < (3pi)/2`

⇒ x is in III quadrant

From ΔPQR,

PQ = `sqrt(5^2 - 4^2)`

= `sqrt(25 - 16)`

= `sqrt(9)`

= 3

Since x is in III quadrant

Both sin x and cos x are negative

∴ sin x = `- 3/5` and cos x = `- 4/5`

sin y = `- 24/25` and y is in III quadrant

Both sin y and cos y are negative

From ΔABC,

BC = `sqrt(25^2 - 24^2)`

= `sqrt(625 - 576)`

= `sqrt(49)`

= 7

So, sin y = `- 24/25` = cos x cos y + sin x si y

= `(- 4/5)(- 7/25) + (-3/5)(- 24/25)`

= `28/125 + 72/125`

= `100/125`

= `4/5`