Question Video: Using Cosine Role to Calculate an Unknown Side Length of a Triangle Mathematics • 11th Grade

Calculate side 𝑏 to the nearest hundredth.

03:58

Video Transcript

Calculate side 𝑏 to the nearest hundredth.

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.