Answer:
Step-by-step explanation:
One angle can be found using the Law of Cosines:
a² = b² +c² -2bc·cos(A)
cos(A) = (b² +c² -a²)/(2bc)
Then the angle opposite the shortest side has the cosine ...
cos(A) = (10² +13² -7²)/(2·10·13) = 220/260 = 11/13
The measure of that angle is ...
A = arccos(11/13) ≈ 32.2042°
The measure of the second-largest angle is then found using the Law of Sines:
sin(B)/sin(A) = b/a
sin(B) = (b/a)sin(A) = 10/7·sin(32.32042°) ≈ 0.761341
B = arcsin(0.761341) ≈ 49.5826°
The remaining angle makes the sum of angles be 180°.
C = 180° -49.5826° -32.2042° = 98.2132°
Rounded to 1 decimal place, the angles opposite the given sides are ...
- side 7 m: 32.2°
- side 10 m: 49.6°
- side 13 m: 98.2°
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Comment on the solution method
The Law of Cosines finds angle values without the ambiguity that is sometimes present when using the law of sines. If it is used to find the largest angle first (opposite the longest side), then that ambiguity is resolved.
We found the smallest two angles first, so did not use the Law of Sines to find the largest angle. That is how we chose to resolve any possible ambiguity here.
