Question Video: Using the Cosine Rule to Find the Side Lengths of a Triangle Physics

Video Transcript

Find the length of side 𝑎 of the triangle shown.

Okay, so in this triangle, we have an interior angle marked out 49 degrees, and opposite that is the side length we want to solve for, 𝑎. To help us do this, we’re given the other two side lengths: 9.4 centimeters and 12 centimeters. And knowing all this about this triangle, we can use what’s called the cosine rule to solve for the side length 𝑎. This rule says that if we have any triangle and we label the interior angles capital 𝐴, 𝐵, and 𝐶 and that if we call the corresponding side lengths lowercase 𝑎, 𝑏, and 𝑐, then the side length 𝑎 squared is equal to the side length 𝑏 squared plus the side length 𝑐 squared minus two times 𝑏 times 𝑐 all multiplied by the cos of the angle 𝐴.

So, this rule is well set up for letting us solve for the unknown side length 𝑎 in our triangle. To solve for that side length, we’ll want to figure out which information in this triangle corresponds to side length 𝑏 and to side length 𝑐 and to angle 𝐴. We can see right away that it’s this 49-degree angle that we’ll call angle 𝐴 for our cosine rule equation. But then, what about our two side lengths? Which one of these two is 𝑏 and which one will we call 𝑐? As we look at our cosine rule equation though, we can see that this choice doesn’t make a mathematical difference. Whichever side length we call 𝑏 and calling the other one 𝑐, we’ll find the same answer for our side length 𝑎.

Just to make a particular choice though, let’s call our side length of 12 centimeters 𝑏. And that means our side length of 9.4 centimeters is 𝑐. Now that we know the side lengths 𝑐 and 𝑏 and the angle 𝐴, we have all the information we need to fill in the right-hand side of this equation. So, we can write then that our 𝑎 squared, where 𝑎 is this unknown side length we want to solve for, is equal to 12 centimeters squared plus 9.4 centimeters squared minus two times 12 centimeters times 9.4 centimeters times the cos of 49 degrees.

Before we enter this expression on our calculator, there’s one last step to take. We want to solve for 𝑎 rather than 𝑎 squared. So let’s take the square root of both sides of this equation. When we do that, on the left-hand side, the power of two cancels out with our square root, which is effectively a power of one-half. And now we have an expression for the side length 𝑎 we want to solve for. When we calculate the right-hand side, to two significant figures, we find a result of 9.2 centimeters. This is the length of side 𝑎 in our triangle.