Video Transcript
𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 is equal to 13.8 centimeters, 𝐴𝐶 is equal to 21.2 centimeters, and the measure of angle 𝐴 is 21 degrees. Find all possible values of length 𝐴𝐵 giving the answer to three decimal places.
We begin by drawing a quick sketch of what the triangle may look like to help us visualize the problem. We are told that sides 𝐵𝐶 and 𝐴𝐶 have lengths equal to 13.8 centimeters and 21.2 centimeters, respectively. The measure of angle 𝐴 is 21 degrees. We have been asked to find all possible values of length 𝐴𝐵. If we knew the measure of angle 𝐶, we could do this using the law of sines. This states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶, where uppercase 𝐴, 𝐵, and 𝐶 are the measures of the three angles and lowercase 𝑎, 𝑏, and 𝑐 are the lengths of the sides opposite them.
In order to calculate the possible measures of angle 𝐶, we begin by calculating angle 𝐵. Using the alternative version of the law of sines where we have the reciprocal of each term, we have sin 𝐵 over 21.2 is equal to sin of 21 degrees over 13.8. We can then multiply both sides of this equation by 21.2 as shown. Typing the right-hand side into our calculator, we have sin 𝐵 is equal to 0.5505 and so on. We then take the inverse sine of both sides of our equation. 𝐵 is therefore equal to 33.4038 and so on. One possible measure of angle 𝐵 to three decimal places is 33.404 degrees.
At this stage, we also need to check for the other angle that also has a sine equal to 0.5505 and so on. Since the sin of 180 degrees minus 𝜃 is equal to sin 𝜃, this is the supplementary angle. The measure of angle 𝐵 could therefore also be equal to 180 minus 33.4038 and so on. This is equal to 146.5961 and so on. Since angles in a triangle sum to 180 degrees and the sum of this angle plus 21 degrees is less than 180, we have two possible valid answers for angle 𝐵, which can be demonstrated in the two triangles shown.
We are now in a position to calculate the two possible measures of angle 𝐶. Subtracting the measures of angle 𝐴 and 𝐵 from 180 degrees, we have 125.596 degrees and 12.404 degrees. We are now in a position to calculate the possible values of length 𝐴𝐵 using the law of sines. Ensuring that we use our nonrounded values for accuracy, we have 𝑐 over sin of 125.596 degrees is equal to 13.8 over sin of 21 degrees. Multiplying through by sin of 125.596 degrees, we have 𝑐 is equal to 31.312 centimeters to three decimal places.
Next, we repeat this process for the second possible measure of angle 𝐶. We have 𝑐 over sin of 12.404 degrees is equal to 13.8 over sin of 21 degrees. Multiplying through by sin of 12.404 degrees, we have 𝑐 is equal to 8.272 centimeters to three decimal places. We can therefore conclude that the two possible lengths of 𝐴𝐵 are 31.312 centimeters and 8.272 centimeters to three decimal places.