# In triangle MNP, m=16, n=15, p=8, how do you find the cosine of each of the angles?

Dec 9, 2016

$\cos \hat{M} = 0.1375$
$\cos \hat{N} = 0.37109375$
$\cos \hat{P} = 0 , 86875$

#### Explanation:

If you know two sides and the included angle of a triangle ABC, you can apply the cosine thorem:

${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cos \hat{A}$

Now you can use the inverse formula to know the cosine of an angle when you know the three sides:

$\cos \hat{A} = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$

Then

$\cos \hat{M} = \frac{{n}^{2} + {p}^{2} - {m}^{2}}{2 n p}$

$= \frac{{15}^{2} + {8}^{2} - {16}^{2}}{2 \cdot 15 \cdot 8}$

$= 0.1375$

hatM=82.1°

$\cos \hat{N} = \frac{{m}^{2} + {p}^{2} - {n}^{2}}{2 m p}$

$= \frac{{16}^{2} + {8}^{2} - {15}^{2}}{2 \cdot 16 \cdot 8}$

$= 0.37109375$

hatN=68.22°

$\cos \hat{P} = \frac{{m}^{2} + {n}^{2} - {p}^{2}}{2 m n}$

$= \frac{{16}^{2} + {15}^{2} - {8}^{2}}{2 \cdot 16 \cdot 15}$

$= 0 , 86875$

hatP=29.69°