# In triangle DEF, d=15, e=12, f=8, how do you find the cosine of each of the angles?

Mar 8, 2017

$\cos D = - \frac{17}{192} , \cos E = \frac{29}{48} , \mathmr{and} , \cos F = \frac{61}{72.}$

#### Explanation:

By the Cosine Rule, $\cos D = \frac{{e}^{2} + {f}^{2} - {d}^{2}}{2 e f}$

$\therefore \cos D = \frac{144 + 64 - 225}{2 \left(12\right) \left(8\right)} = - \frac{17}{192.}$

$\cos E = \frac{{f}^{2} + {d}^{2} - {e}^{2}}{2 f d} = \frac{64 + 225 - 144}{2 \left(8\right) \left(15\right)} = \frac{145}{2 \left(8\right) \left(15\right)} = \frac{29}{48.}$

$\cos F = {d}^{2} + {e}^{2} - {f}^{2} / \left(2 \mathrm{de}\right) = \frac{225 + 144 - 64}{2 \left(15\right) \left(12\right)} = \frac{305}{2 \left(15\right) \left(12\right)} = \frac{61}{72.}$