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

Video Transcript

Find the size of angle 𝐴, in degrees, of the triangle shown.

Looking at this triangle, we see that all three side lengths are given to us, and we want to solve for this unknown angle here. Given this information, we can use the cosine rule to solve for this angle 𝐴. The cosine rule tells us that given a triangle with interior angles capital 𝐴, 𝐵, and 𝐶 and corresponding side lengths lowercase 𝑎, 𝑏, and 𝑐, then if we pick out the particular side length we’ve labeled 𝑎, that length 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.

As we look to apply the cosine rule to our scenario though, we can see it’s not any side length we want to solve for but rather an interior angle. We can say then that in the cosine rule equation, it’s this angle here that we want to solve for. And we can do that by rearranging this expression. As a first step to doing that, we can subtract 𝑏 squared and 𝑐 squared from both sides. This means that on the right-hand side, we have positive 𝑏 squared minus 𝑏 squared canceling that out and positive 𝑐 squared minus 𝑐 squared canceling that out as well.

Next, we’ll continue to move toward isolating this angle 𝐴 by dividing both sides of the equation by negative two times 𝑏 times 𝑐, which means on the right-hand side, this factor of negative two times 𝑏 times 𝑐 cancels. That leaves us with this expression. And the last thing we’ll do to get this angle 𝐴 all by itself is to invert our operation of the cosine function by taking the inverse cosine or arc cosine of both sides. On the right-hand side, when we take the inverse cos of the cos of the angle 𝐴, those inverse cosine and cosine functions effectively cancel out. So now, we have an expression letting us solve for the angle 𝐴.

Notice that to do this, we need to know all three of the side lengths of our triangle, lowercase 𝑎, 𝑏, and 𝑐. And as we look over at the particular triangle on this example, we see we do have that information. We can say that 7.8 centimeters is our side length lowercase 𝑎. And although it doesn’t make a mathematical difference whether we call this side length 𝑏 or this one 𝑏, which would mean that the remaining side length we call side length 𝑐, for the sake of clearly identifying our values, let’s call this 14-centimeter side length side length 𝑏. And that means 9.6 centimeters is 𝑐.

And now we can insert these side lengths into an equation to solve for the angle 𝐴. It’s equal to the inverse cos of this whole argument where we’ve plugged in values for side lengths 𝑎, 𝑏, and 𝑐, respectively. Notice, by the way, that all our side lengths have the same units, centimeters. Since they’re all consistent on that basis, we can go ahead and calculate the right-hand side of this equation. Rounding our answer to two significant figures, we find a result of 32 degrees. This is the size of the angle 𝐴 in this triangle.