Question Video: Using the Cosine Rule to Find the Measures of a Triangle’s Three Angles

Video Transcript

For the triangle given, find the value of the three angles to the nearest degree.

In the figure, we are given the lengths of all three sides of our triangle. Side length 𝐴 opposite angle 𝐴 is equal to 27 centimeters. Side length 𝐵 is equal to 28 centimeters. And side length 𝐶 is equal to nine centimeters. We need to calculate the measure of angle 𝐴, angle 𝐵, and angle 𝐶. We can do this using the law of cosines, which states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by the cos of angle 𝐴. The lowercase letters here correspond to the side lengths and the capital or uppercase 𝐴 is an angle.

When we wish to calculate the measure of an angle as opposed to the length of a side, we can use a rearrangement of this formula. This states that the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two 𝑏𝑐. We will begin by substituting our values from the figure to calculate the measure of angle 𝐴. The cos of angle 𝐴 is equal to 28 squared plus nine squared minus 27 squared all divided by two multiplied by 28 multiplied by nine. Typing the right-hand side into our calculator gives us 17 over 63. We can then take the inverse cos of both sides of this equation such that angle 𝐴 is equal to the inverse cos of 17 over 63. Typing this into the calculator gives us 74.3451 and so on.

We are asked to give the value of the three angles to the nearest degree. We can therefore conclude that the measure of angle 𝐴 to the nearest degree is 74 degrees. We can now repeat this process to calculate the measure of angle 𝐵. The cos of angle 𝐵 is equal to 𝑎 squared plus 𝑐 squared minus 𝑏 squared all divided by two 𝑎𝑐. In this question, we have 27 squared plus nine squared minus 28 squared all divided by two multiplied by 27 multiplied by nine. This time, the right-hand side simplifies to 13 over 243. Angle 𝐵 is equal to the inverse cos of 13 over 243. Typing this into the calculator gives us 86.9333 and so on. Once again, we need to round to the nearest degree such that the measure of angle 𝐵 is 87 degrees.

Finally, we can repeat this process to calculate the measure of angle 𝐶. The cos of angle 𝐶 is equal to 27 squared plus 28 squared minus nine squared all divided by two multiplied by 27 multiplied by 28. The right-hand side of this equation is equal to 179 over 189. Therefore, angle 𝐶 is equal to the inverse cos of 179 over 189. Rounding our value of 18.7214 and so on to the nearest degree, we see that the measure of angle 𝐶 is 19 degrees.

At this stage, it is worth recalling that the three angles in a triangle must sum to 180 degrees. Whilst we need to use the exact values to check this, adding 74, 87, and 19 will give us a good indication as to whether our answers are correct. In this case, the three angles do sum to 180 degrees. In general, provided we are close to 180, our answers are likely to be correct. In this question, the measure of angle 𝐴 is 74 degrees, angle 𝐵 is 87 degrees, and angle 𝐶 is 19 degrees, where all three are measured to the nearest degree.