WAEC 2008 · Paper 2 · Q2

(a)
If $\sin A = \frac35$ and $\cos B = \frac{15}{17}$ , where $A$ is obtuse and $B$ is acute, find the value of $\cos(A + B)$ .

Worked solution (try it first)

Use $\sin^2 A + \cos^2 A = 1$ : $\cos^2 A = 1 - \frac{9}{25}$
$= \frac{16}{25}$ .
$A$ is obtuse, so its cosine is negative: $\cos A = -\frac45$ .
For $B$ : $\sin^2 B = 1 - \frac{225}{289}$
$= \frac{64}{289}$ .
$B$ is acute, so $\sin B = \frac{8}{17}$ .
Use $\cos(A + B) = \cos A \cos B - \sin A \sin B$ .
Substitute: $\left(-\frac45\right)\left(\frac{15}{17}\right) - \left(\frac35\right)\left(\frac{8}{17}\right) = -\frac{60}{85} - \frac{24}{85}$ .
$\cos(A + B) = -\frac{84}{85}$ .