Question

Suppose that the terminal side of angle alphaα lies in Quadrant I and the terminal side of angle betaβ lies in Quadrant IV. If sine alpha equals five thirteenthssinα= 5 13 and cosine beta equals StartFraction 6 Over StartRoot 85 EndRoot EndFractioncosβ= 6 85​, find the exact value of cosine left parenthesis alpha plus beta right parenthesiscos(α+β).

Answer 1

It is given that :

`$\alpha$` lies in the first quadrant.

And
`$\beta$` lies in the fourth quadrant.

Since,
`$\sin \alpha = (5)/(13)$` and
`$\cos \beta = (6)/(√(85))$` (given)

`$\sin \alpha = (5)/(13)$`

`$\cos \alpha = √(1-\sin^2 \alpha)$`

`$\cos \alpha = (12)/(13)$`

Similarly
`$\cos \beta = (6)/(√(85))$`

`$\sin \beta = √(1-\cos^2 \beta)$`

`$\sin \beta = \sqrt{1-(36)/(85)}$`

`$-(7)/(√(85))$` (IVth quadrant)

Therefore,

`$\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$`

`$=(12)/(13)* (6)/(√(85))-(5)/(13)* (-7)/(√(85))$`

`$= (107)/(13 √(85))$`