# 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°
The text passage contains a mathematical question about finding the cosine of each angle in a triangle given the lengths of its sides, and it provides answers for the cosines of the angles.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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°
user
The text passage contains a mathematical question about finding the cosine of each angle in a triangle given the lengths of its sides, and it provides answers for the cosines of the angles.

TYPE: SINGLE_QUESTION_WITH_ANSWER