Question #a69e0

1 Answer
Nov 18, 2016

Answer:

#A~~44.2^@#
#B ~~ 95.5^@#
#C ~~ 40.3^@#

Explanation:

By convention, #a# represents the side opposite vertex #A#, #b# represents the side opposite vertex #B#, and #c# represents the side opposite vertex #C#.

The law of cosines states that #c^2 = a^2+b^2-2abcos(C)#. Note that this is true regardless of how the sides and vertices are labeled, so it is also true that

  • #b^2 = a^2+c^2-2ac cos(B)#
  • #a^2 = b^2+c^2-2bc cos(A)#

If we try solving for the angle, we get

  • #cos(A) = (b^2+c^2-a^2)/(2bc)#
  • #cos(B) = (a^2+c^2-b^2)/(2ac)#
  • #cos(C) = (a^2+b^2-c^2)/(2ab)#

Thus, applying the inverse cosine function to both sides:

  • #A = arccos((b^2+c^2-a^2)/(2bc))#
  • #B = arccos((a^2+c^2-b^2)/(2ac))#
  • #C = arccos((a^2+b^2-c^2)/(2ab))#

Substituting in the given values #a=14, b=20, c=13# into a calculator, we get our results:

  • #A~~44.2^@#
  • #B ~~ 95.5^@#
  • #C ~~ 40.3^@#