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

Solution :

It is given that :

$\alpha$ lies in the first quadrant.

And $\beta$ lies in the fourth quadrant.

Since, $\sin \alpha = \frac{5}{13}$     and $\cos \beta = \frac{6}{\sqrt{85}}$    (given)

$\sin \alpha = \frac{5}{13}$  

$\cos \alpha = \sqrt{1-\sin^2 \alpha}$

   $\cos \alpha = \frac{12}{13}$

Similarly  $\cos \beta = \frac{6}{\sqrt{85}}$

$\sin \beta = \sqrt{1-\cos^2 \beta}$

$\sin \beta = \sqrt{1-\frac{36}{85}}$

     $-\frac{7}{\sqrt{85}}$      (IVth quadrant)

Therefore,

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

                 $=\frac{12}{13}\times \frac{6}{\sqrt{85}}-\frac{5}{13}\times \frac{-7}{\sqrt{85}}$

                $= \frac{107}{13 \sqrt{85}}$