Let's look at word problems that use a law of cosines.

Two airplanes leave an airport at the same time.

An hour later, they were are 189 kilometers apart.

If one plane traveled 168 kilometers,

and the other plane traveled 244 kilometers during that hour,

find the angle theta between their flight paths.

Now, the figure up here shows the situation.

Let's label the angles and sides with a standard law of cosine notation.

In other words, let's let this angle theta that we're looking for be capital A.

Let this angle of this plane over here be capital B.

We'll let this angle of this plane here be capital C. Now remember,

we label the sides opposite the angle with the corresponding letter but just lower case.

So this is a little c, this is little a,

and this is little b.

So we're looking for angle theta which is angle A.

So the following law of cosine formula can help us.

A squared is equal to b squared plus c squared minus two,

times b, times c,

times the cosine of A,

and plugging in our information gives us 189 squared is equal to 168,

plus 244 squared, minus two,

times 168, times 244,

times the cosine of A.

And now, let's bring this entire product here to

the left hand side and bring this to the right hand side,

which gives us two times 168, times to 244,

times cosine of A is equal to 168 squared,

plus 244 squared, minus 189 squared.

And now, dividing both sides by this product here gives us that cosine of A,

is equal to 168 squared, plus 244 squared,

minus 189 squared, divided by,

2 times 168, times 244.

Which means that A is the angle whose cosine is this ratio here.

But moreover, since A is an angle on a triangle,

it has to be between zero and 180 degrees.

That is A is equal to inverse cosine of this whole ratio,

168 squared, plus 244 squared,

minus 189 squared, divided by,

2 times 168, times 244.

And plugging this in our calculator we get that this is approximately 50.6 degrees.

And looking back up here in our figure then,

remember A is theta which is what we were looking for.

So theta then our answer is approximately,

50.6 degrees and this

is how we can use the law of cosines to help us solve a word problem.

Thank you, and we'll see you next time.