Suppose a triangle has two sides of length 32 and 35, and that the angle between these two sides is 120. What is the length of the third side of the triangle?
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You have the triangle in the figure above.
The law of sines is easier to use than the law of cosines, so whenever you can use the law of sines you should.
If you can't use the law of sines, then you use the law of cosines.
How do you know if you can use the law of sines? You can use the law of sines if you know the measure of an angle and the length of the opposite side. You must have one angle and its opposite side to establish the ratio of the law of sines.
In this problem, we have an angle, but not the opposite side. Then we have two sides but not their opposite angles. Since there is no angle and its opposite sides known, you can't use the law of sines. That means you must use the law of cosines.
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The law of cosines can be written in three different ways.
With the given information you have, this is the version you need.
Using it, you can solve for a.
Once you find a, then you can use the law of sines to find angle B or C.
Once you have angle B or C, then you can subtract the two known angles form 180 to find the third angle.
\(\large a^2 = b^2 + c^2 - 2bc \cos A\)