Find the exact value of the trigonometric function $$\cos(u-v),$$ given that $\sin(u)=\frac{12}{13}$ and $\cos(v)=-\frac{4}{5}$ (both $u$ and $v$ are in quadrant II).
$\endgroup$ 21 Answer
$\begingroup$$$\cos(u-v) = \cos(u)\cos(v) + \sin(u)\sin(v)$$
Using the fact that
$$\cos^2(x) + \sin^2(x) = 1,$$
we can see that $\cos(u) = \pm 5/13$ and $\sin(v) = \pm 3/5$. Since the two angles are in quadrant 2 we can say that $\cos(u) = -5/13$ and $\sin(v) = +3/5$.
Then,
$$\cos(u-v) = -\frac{5}{13}\left(-\frac{4}{5}\right) + \frac{12}{13}\left(\frac{3}{5}\right)=\frac{56}{65}.$$
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