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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`
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Solution
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`
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