Professor Dave again, let’s learn the law of cosines.
We just learned the law of sines for oblique triangles, so let’s learn one more law of
importance.
That’s the law of cosines.
This one is sort of like the Pythagorean theorem, in that it relates the squares of the lengths
of the triangle, but the Pythagorean theorem works only for right triangles, whereas the
law of cosines works for any triangle.
Here is a general triangle with sides of length A, B, and C, in lower case, and angles A,
B, and C in upper case which correspond to the lengths directly opposite.
What this law says is that if we know the length of any two sides of a triangle, and
the measure of the angle between them, we can calculate the length of the third side.
A squared equals B squared plus C squared minus two BC cosine A. As you can see, the
square of the missing length is expressed in terms of the other two lengths and the
angle between them.
While the law of sines allowed us to solve side angle angle, angle side angle, and side
side angle triangles, now with the law of cosines we can solve side angle side triangles.
Take this one for example.
We’ve got our two sides and the angle between them known, so let’s plug everything into
the equation, simplify a little bit, and there’s our value for the third side.
From here, we could use the law of sines to get one more angle, and then subtract these
two from one eighty to get the final angle.
Now we know everything about this triangle.
Notice that we couldn’t have solved this with the law of sines alone, because to use
that law we need one side and angle that correspond to one another, as well as one other piece
of information, in order to start using the ratio.
With the law of cosines, we can also solve for side side side triangles.
This is because if we know all three side lengths, we can solve for any one of the angles.
The easiest way to do this is to solve for the angle that is opposite the longest side,
so for this triangle, let’s call the longest side C. Using the general formula, let’s
solve for angle C. Bring this term containing the angle to the other side to make it positive,
and bring A squared to the other side.
Then we just divide, and now we are ready to plug in all the values.
We plug in side lengths A, B, and C, simplify, and we get our answer for cosine of A. We
take the inverse cosine, and there’s our angle to the nearest tenth.
From here, we just use the law of sines to get one more angle, and then subtract these
two from one eighty to get the third angle.
So as we can see both the law of sines and law of cosines are very straightforward in
their application, so let’s go ahead and check comprehension.