text: # 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°
explanation: The document contains a math question about finding the cosine of each angle in a triangle given the side lengths. It also provides the answers to the question, including the cosine values for each angle and the angles themselves.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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°
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question about finding the cosine of each angle in a triangle given the side lengths. It also provides the answers to the question, including the cosine values for each angle and the angles themselves.
  
  Label: question-and-answer