Question Video: Using the Law of Cosines to Find Unknown Angles and Lengths of a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where 𝑎 equals 28 centimeters, 𝑏 equals 17 centimeters, and the measure of angle 𝐶 equals 60 degrees. Find the missing length rounded to three decimal places and the missing angles rounded to the nearest degree.

Let’s start by sketching this triangle and adding the information we know. Here, we have a triangle, and we can label it with the vertices 𝐴, 𝐵, and 𝐶. Conventionally, the side length opposite the vertex 𝐴 is labeled with a lowercase 𝑎. The side length opposite vertex 𝐵 would be labeled with the lowercase 𝑏. And the side length opposite vertex 𝐶 would be labeled with the lowercase 𝑐.

We know that 𝑎 equals 28 centimeters, 𝑏 equals 17 centimeters, and angle 𝐶 measures 60 degrees. And we want to solve for the measures of angle 𝐴 and 𝐵. And we want to solve for the measures of angle 𝐴 and 𝐵 and the length of side length 𝑐. We know two of the sides and the angle in between them. We have enough information to use the law of cosines to solve for that third missing side. The law of cosines tells us that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 times cos of 𝐴.

Let’s think about what this means. If we know some angle 𝐴 inside a triangle and the two side lengths adjacent to angle 𝐴, we have enough information to find the side length of the side that is opposite to our angle 𝐴. In our triangle, we’re actually missing the side length 𝑐 and we know angle 𝐶. Because we know 𝐴 and 𝐵 and angle 𝐶, it might be helpful to rewrite this law in terms of the missing side length 𝑐. Since we’re looking for side length 𝑐, we would say that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 times cos of 𝐶. And then to solve for 𝑐, we just plug in what we know. 𝑐 squared will be equal to 28 squared plus 17 squared minus two times 28 times 17 times cos of 60.

Calculating that, we end up with 𝑐 squared equals 579. We’ll find 𝑐 by taking the square root of both sides. And we find that 𝑐 equals 24.43358345 continuing. We know that this side length needs to be rounded to three decimal places. To the right of that three is a five. And so we can round the side length 𝑐 to 24.434 centimeters. And we’ll add that value to our graph.

At this point, we want to find a missing angle in this triangle. If we look closely, we’ll see that we can rearrange the law of cosines to use it to solve for a missing angle if we know all three side lengths in a triangle. If we knew side lengths 𝑎, 𝑏, and 𝑐, the only variable remaining is that opposite angle 𝐴. It’s possible to go ahead and plug in all the information as it’s written here. However, we could rewrite the equation so that cos of 𝐴 is by itself.

If we subtract 𝑏 squared and 𝑐 squared from both sides of the equation, we end up with 𝑎 squared minus 𝑏 squared minus 𝑐 squared equals negative two 𝑏𝑐 times cos of 𝐴. So we divide both sides of the equation by negative two 𝑏𝑐. Negative two 𝑏𝑐 divided by negative two 𝑏𝑐 just leaves cos of 𝐴. And so cos of 𝐴 is equal to 𝑎 squared minus 𝑏 squared minus 𝑐 squared over negative two 𝑏𝑐, substituting in the values we know for 𝑎, 𝑏, and 𝑐 and then using a calculator to calculate this. To do that accurately, make sure you put some form of brackets or parentheses around your numerator and your denominator. When we do that, we get the cos of 𝐴 equals 0.1228042 continuing.

Since we’re interested in the measure of this angle, we need to take the inverse cosine of both sides of this equation. This tells us that the measure of angle 𝐴 is 82.9460 continuing degrees. We need to round this to the nearest degree. So we’ll say that 𝐴 equals 83 degrees. And then we’ll add that value to our diagram. Of course, we could use the law of cosines a third time to find the measure of angle 𝐵. However, because we’re working inside a triangle, we know that the three angles must sum together to be 180 degrees. And so we can solve for the measure of angle 𝐵 by subtracting 83 degrees and 60 degrees from 180 degrees. When we do that, we find the measure of angle 𝐵 is equal to 37 degrees.

And so we found our three unknown values for this triangle. Side length 𝑐 equals 24.434 centimetres, measure of angle 𝐴 is 83 degrees, and the measure of angle 𝐵 is 37 degrees.