SOLUTION: the side of a triangle measure 16 centimeters, 13 centimeters, and 11 centimeters, what are the measures of the angles of the triangle

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Question 1001510: the side of a triangle measure 16 centimeters, 13 centimeters, and 11 centimeters, what are the measures of the angles of the triangle
Answer by KMST(5328) About Me  (Show Source):
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In a right triangle the longest side is the hypotenuse,
and if a= length of the hypotenuse, with b and c being the other sides lengths,
a%5E2=b%5E2%2Bc%5E2 (the Pythagorean relationship).
In any triangle ABC,
a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 (the law of cosines).
(We can also write it as b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29 or c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29 ).
So if a=16= length of side opposite angle A
16%5E2=13%5E3%2B11%5E2-2%2A13%2A11%2Acos%28A%29
256=169%2B121-286%2Acos%28A%29
286%2Acos%28A%29=169%2B121-256
286%2Acos%28A%29=34
cos%28A%29=34%2F286--->A=about1.4516 or A=about83.17%5Eo (rounded)
NOTE:
We can also write the law of cosines as
b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29 or c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29 .

To find the measure of another angle,
we can apply the law of cosines again,
or we can use the law of sines.
After that, we find the remaining angle by using the fact that the measures of all 3 angles add up to pi or 180%5Eo .

USING LAW OF COSINES AGAIN:
let B be the angle opposite the side with length b=13cm.
13%5E2=16%5E2%2B11%5E2-2%2A16%2A11%2Acos%28B%29
169=256%2B121-352%2Acos%28B%29
352%2Acos%28B%29=256%2B121-169
352%2Acos%28B%29=208
cos%28B%29=208%2F352--->B=about0.9386 or B=about53.78%5Eo (rounded).

USING LAW OF SINES FOR A CHANGE:
If you have the measures of one side and the opposite angle of a triangle,
you can apply the law of sines, which states
sin%28B%29%2Fb=sin%28A%29%2Fa=sin%28C%29%2Fc
So, with b=13cm,
sin%28B%29%2F13=sin%28A%29%2F16--->sin%28B%29=13sin%28A%29%2F16
sin%28A%29=about0.9929 (rounded), so
sin%28B%29=about13%2A0.9929%2F16--->B=about0.9386 or B=about53.78%5Eo (rounded).

Once we have system%28A=about1.4516%2CB=about0.9386%29 or system%28A=about83.17%5Eo%2CB=about53.78%5Eo%29
We can find
C=pi-about1.4516-about0.9386=about0.7514 or
C=180%5Eo-83.17%5Eo-about53.78%5Eo=about43.05%5Eo .