Question 622031: please help me solve this triangle:
side a= 6, side b= 9, side c= 14
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! With 3 sides and no angles, you cannot use the Law of Sines (yet). We must start by using the Law of Cosines. And here's a tip, with three sides, use the Law of Cosines to find the largest angle first. (This is helpful because only the largest angle can be an obtuse or right angle. After you find the largest angle, we can freely use the simpler formula, Law of Sines, to find the second angle without having to wonder if that angle is obtuse (or right)).
The largest angle, as always, will be opposite the largest side. So it will be angle C. So we will use
Inserting the values we have for the sides we get:
Simplifying...
196 = 36 + 81 - 108cos(C)
196 = 117 - 108cos(C)
Now we solve for cos(C). Subtracting 117:
85 = -180cos(C)
Dividing by -108:
-0.78703704 = cos(C)
(With a negative value here we can see that we are indeed going to have an obtuse angle.)
Using the inverse cos button on our calculator:
141.9 = C
Now we can use the Law of Cosines again or the Law of Sines to find the next angle. Since the Law of Sines is a simpler formula I'm going to use it. Which angle we find doesn't matter. Let's find A using
Inserting the values we have we get:
Now we solve for sin(A). Multiplying by 6:
sin(A) = 0.26438818
Now using the inverse sin button:
A = 15.3
(Note: Because we already found the largest angle, we do not have to wonder if A is 15.3 or 180-15.3 degrees.)
For the last angle, B, we can use the Law of Cosines, Law of Sines, or ... we can just use the fact that the three angles must add up to 180:
B = 180 - (141.9 + 15.3)
B = 22.8
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