### Video Transcript

𝐴𝐵𝐶 is a triangle, where the length of side 𝑎 is 13.8 centimeters, the length of side 𝑏 is 15.9 centimeters, and the measure of angle 𝐴 is 28 degrees. Find all possible values for the other lengths and angles, giving lengths to two decimal places and angles to the nearest second.

We know that one way of calculating missing lengths and angles in triangles is using the sine rule or 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 the capital letters correspond to the angles and the lowercase letters are the lengths of the sides opposite these. In this question, we are told that the side lengths 𝑎 and 𝑏 are 13.8 and 15.9 centimeters, respectively. This means we need to calculate side length 𝑐. We are also told that the measure of angle 𝐴 is 28 degrees. This means that we need to calculate the measure of angles 𝐵 and 𝐶.

We will begin by trying to calculate the measure of angle 𝐵. Substituting in our values, we have 13.8 over sin of 28 degrees is equal to 15.9 over the sin of angle 𝐵. We can cross multiply here such that 13.8 multiplied by the sin of angle 𝐵 is equal to 15.9 multiplied by the sin of 28 degrees. We can then divide both sides of this equation by 13.8. We can then take the inverse or arcsine of both sides of this equation. Angle 𝐵 is equal to 32.7458 and so on degrees. We need to give our angles to the nearest second. Using the degrees, minutes, and seconds button on our calculator gives us 32 degrees, 44 minutes, and 45 seconds. This is a possible value for the measure of angle 𝐵.

We know that the three angles in any triangle sum to 180 degrees. This means that 28 degrees plus 32 degrees, 44 minutes, and 45 seconds plus angle 𝐶 must equal 180 degrees. Adding the measure of angle 𝐴 and 𝐵, we can then subtract 60 degrees, 44 minutes, and 45 seconds from both sides of our equation. Angle 𝐶 is therefore equal to 119 degrees, 15 minutes, and 15 seconds. When the measure of angle 𝐵 is 32 degrees, 44 minutes, and 45 seconds, the measure of angle 𝐶 is 119 degrees, 15 minutes, and 15 seconds.

We can now use the sine rule once again to calculate the length of side 𝑐. 13.8 divided by sin of 28 degrees is equal to 𝑐 divided by sin of 119 degrees, 15 minutes, and 15 seconds. We can then multiply both sides of the equation by sin of 119 degrees, 15 minutes, and 15 seconds. This gives us a value of 𝑐 equal to 25.6457 and so on. The length of side 𝑐 is 25.65 centimeters to two decimal places. This gives us one possible set of values for the measure of angle 𝐵, the measure of angle 𝐶, and the length of side 𝑐.

We might think that this is the end of this question. However, when dealing with a problem of this type, we need to recall our sine graph or CAST diagram. This tells us that there will be two angles between zero and 180 degrees whose sine give the same value. We can calculate this second value by subtracting the measure of angle 𝐵 we found earlier from 180 degrees. This gives us 147 degrees, 15 minutes, and 15 seconds. This is a second possible value for the measure of angle 𝐵.

We can now once again consider that angles in a triangle sum to 180 degrees. 28 degrees plus 147 degrees, 15 minutes, and 15 seconds plus angle 𝐶 must equal 180 degrees. We can therefore calculate the measure of angle 𝐶 by subtracting 175 degrees, 15 minutes, and 15 seconds from 180 degrees. This gives us four degrees, 44 minutes, and 45 seconds. When the measure of angle 𝐵 is 147 degrees, 15 minutes, and 15 seconds, then the measure of angle 𝐶 is four degrees, 44 minutes, and 45 seconds.

We will now use the sine rule once again to calculate the corresponding value of side length 𝑐. 13.8 divided by the sin of 28 degrees is equal to 𝑐 divided by the sin of four degrees, 44 minutes, and 45 seconds. We can rearrange this equation to calculate the value of 𝑐. 𝑐 is equal to 2.4319 and so on. Rounding this to two decimal places gives us 2.43 centimeters.

We have two possible sets of values for the missing lengths and angles. Either angle 𝐵 is equal to 32 degrees, 44 minutes, and 45 seconds; angle 𝐶 is equal to 119 degrees, 15 minutes, and 15 seconds; and side length 𝑐 is equal to 25.65 centimeters. Alternatively, angle 𝐵 could be equal to 147 degrees, 15 minutes, and 15 seconds; angle 𝐶 equal to four degrees, 44 minutes, and 45 seconds; and the length of side length 𝑐 equal to 2.43 centimeters.