Question

Find the exact value of cos​(alpha minus beta​) if sin alpha equals 0.5276 and cos beta equals 0.3488​, and both alpha and beta are​ first-quadrant angles?

Answer 1

The exact value:
`cos​(alpha - beta​) = sqrt 0.72163824 xx 0.3488 + 0.5276 xx sqrt 0.87833856`

To four digits: `cos​(alpha - beta​) ≈ 0.7908`

Explanation:

`cos​(alpha - beta​)`
`sin alpha = 0.5276`
`cos beta =0.3488​`

`sin^2 α + cos^2 α = 1`
`0.5276^2 + cos^2 α = 1`
`cos^2 α = 1 - 0.5276^2`
`cos α = + sqrt 0.72163824`

`sin^2 β + cos^2 β = 1`
`sin^2 β + 0.3488^2 = 1`
`sin^2 β = 1 - 0.3488^2`
`sin^2 β = 0.87833856`
`sin β = +sqrt 0.87833856`

`cos​(alpha - beta​) = cosα cosβ+sinα sinβ`
`cos​(alpha - beta​) = sqrt 0.72163824 xx 0.3488 + 0.5276 xx sqrt 0.87833856`