Calculate side 𝑏 to the
Now, within this question,
we’re going to be using the law of cosines. So, I’ll just recall it using
its standard definition if you were to look it up in a textbook. And it’s this definition
here. Now, this is actually gonna be
quite confusing in this example, because we’re not asked to calculate side 𝑎,
we’re asked to calculate side 𝑏. Now, you may think, oh, that’s
okay, I’ll just rearrange this formula so that I get 𝑏 squared equals.
And if you do that, you would
have 𝑏 squared is equal to 𝑎 squared minus 𝑐 squared plus two 𝑏𝑐 cos
𝐴. Now, here’s the problem. This would require you to know
sides 𝑎 and 𝑐, which we do. They’re the sides opposite
angles 𝐴 and 𝐶. So, they’re nine centimetres
and five centimetres. But the other piece of
information we would need is angle 𝐴. And looking at the diagram, you
can see that we haven’t been given angle 𝐴. We’ve been given angle 𝐵.
So, just rearranging the law of
cosines from this standard form doesn’t work because we haven’t been given the
right set of information in order to apply it. Instead, what we need to do is
write our own version of the law of cosines, where we cycle the letters around
so that we’re looking to calculate side 𝑏. So, here is the information we
have, two sides and the included angle, which is exactly the setup that we need
in order to use the law of cosines.
So, what I’m gonna do is I’m
gonna write out the law of cosines again, but cycling the letters through. I want to calculate 𝑏, so I’m
gonna begin with 𝑏 squared. Then, the law of cosines tells
me that I square each of the other two sides. So, in this instance, that’s
going to be 𝑎 squared plus 𝑐 squared. It then tells me that I do
negative two multiplied by 𝑏 and 𝑐, which are the other two sides. Well, in this case, that’s
gonna be negative two multiplied by 𝑎 and 𝑐. Finally, then, I do cos of the
included angle, so in this case that’s going to be cos of angle 𝐵.
So, this isn’t a rearrangement
of the law of cosines because you can see it includes cos of angle 𝐵 instead of
cos of angle 𝐴. Instead, it’s a rewriting of
the law of cosines using the letters in a different order. And now, I have a version that
I can use in order to answer this question. So, I can substitute the
relevant information. I have then that 𝑏 squared is
equal to nine squared plus five squared, first of all, minus two times nine
times five times cos of 120. And now, I can just work
through the stages here.
So, I have 𝑏 squared is equal
to 106 minus 90 cos 120. This tells me that 𝑏 squared
is equal to 151 exactly. That’s because cos of 120 is an
exact value. It’s just negative a half. If I then take the square root,
I have that 𝑏 is equal to 12.288205. And the question has asked me
for this value to the nearest hundredth, so I’ll round my answer. And we have then that 𝑏 is
equal to 12.29 centimetres.
So, when answering a question
like this, you can look up the law of cosines in the form it’s usually given in
or perhaps you’ve committed that form to memory. But if the side you’re looking
for isn’t side 𝑎, then you need to think about how you can swap the letters
around in order to make it relevant for the side you’re looking to
calculate. Remember, you do this by just
considering the structure of the law of cosines and the fact that it includes
the two other sides of the triangle and the included angle, which is the angle
opposite the side you’re looking to calculate.