Question Video: Using the Law of Sines to Calculate Unknown Lengths and Angles of a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where the measure of angle 𝐴 is 57 degrees, six minutes, and 24 seconds; the measure of angle 𝐵 is 31 degrees, 19 minutes, and nine seconds; and length 𝑎 is 21 centimeters. Find all the possible values for the other lengths and angles in triangle 𝐴𝐵𝐶, giving lengths to the nearest centimeter and angles to the nearest second.

Let’s begin by sketching triangle 𝐴𝐵𝐶 as shown. We are told that the measure of angle 𝐴 is 57 degrees, six minutes, and 24 seconds. The measure of angle 𝐵 is 31 degrees, 19 minutes, and nine seconds. And the length of side 𝑎, which is opposite angle 𝐴, is 21 centimeters. We are asked to find all possible values for the other lengths and angles in the triangle. We will begin by finding the measure of angle 𝐶. And we will use this information to help calculate the lengths of side 𝑏 and 𝑐.

We know that angles in a triangle sum to 180 degrees. This means that the measure of angle 𝐶 is equal to 180 degrees minus the sum of the measures of angle 𝐴 and 𝐵. Using the degrees, minutes, and seconds button on our calculator, we find that the sum of angles 𝐴 and 𝐵 is 88 degrees, 25 minutes, and 33 seconds. We can then subtract this from 180 degrees to calculate the measure of angle 𝐶. This is equal to 91 degrees, 34 minutes, and 27 seconds.

To calculate lengths 𝑏 and 𝑐, we will use the fact that we know the measures of two angles and the other side length in our triangle. We will use the sine rule, which states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. Substituting in the values we know, we have 21 over sin 57 degrees, six minutes, and 24 seconds is equal to 𝑏 over sin 31 degrees, 19 minutes, nine seconds is equal to 𝑐 over sin 91 degrees, 34 minutes, 27 seconds.

We will begin by using the first two expressions to calculate length 𝑏. This is equal to 21 over sin 57 degrees, six minutes, 24 seconds multiplied by sin 31 degrees, 19 minutes, nine seconds. Typing this into the calculator gives us 13.000 and so on, which to the nearest centimeter is 13 centimeters. We can repeat this process to calculate side length 𝑐. This is equal to 21 over sin of 57 degrees, six minutes, and 24 seconds multiplied by sin of 91 degrees, 34 minutes, and 27 seconds. Typing this into the calculator gives us 24.999 and so on. To the nearest centimeter, this is equal to 25 centimeters.

Whilst we were asked to give all the possible values for the other lengths and angles, in this question, there is only one possible value for each one. The missing lengths and angles in triangle 𝐴𝐵𝐶 are 𝑏 equals 13 centimeters, 𝑐 equals 25 centimeters, and the measure of angle 𝐶 is 91 degrees, 34 minutes, and 27 seconds.