Question Video: Using the Law of Sines to Calculate All Possible Measures of an Angle in a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle in which 𝑎 is equal to five centimeters, 𝑏 is equal to eight centimeters, and the measure of angle 𝐴 is equal to 36 degrees. If the triangle exists, find all the possible values of angle 𝐵 to the nearest second.

In this question, we are told that the measure of angle 𝐴 in triangle 𝐴𝐵𝐶 is equal to 36 degrees. We will begin by drawing a sketch of what the triangle may look like. We are told that side length 𝑏 is equal to eight centimeters and side length 𝑎 is equal to five centimeters. So if we place a compass set to five centimeters at point 𝐶 and draw an arc, it would intersect the base 𝐴𝐵 in two places. This means that vertex 𝐵 could lie at the point labeled 𝐵 or 𝐵 prime, as shown. And we have two possible triangles in blue and orange where angle 𝐵 has two measures, as shown.

An alternative method to show that there are two possible triangles would be to calculate the value of the height ℎ. We can do this using our knowledge of right angle trigonometry and the sine ratio. We recall that sin 𝜃 is equal to the opposite over the hypotenuse. In our right triangle, sin of 36 degrees is equal to the height ℎ over eight. Multiplying through by eight, we have ℎ is equal to eight multiplied by sin of 36 degrees. This is equal to 4.702 and so on. And to one decimal place, the height of our triangle is 4.7 centimeters.

This is less than the length of side 𝑎 and leads us to a general rule. If angle 𝐴 is acute and the height ℎ is less than side length 𝑎, which is less than side length 𝑏, then two possible triangles 𝐴𝐵𝐶 exist.

We are now in a position to find the two possible values of angle 𝐵 using the law of sines. This states that sin 𝐴 over 𝑎 is equal to sin 𝐵 over 𝑏, which is equal to sin 𝐶 over 𝑐, where uppercase 𝐴, 𝐵, and 𝐶 are the measures of the three angles and lowercase 𝑎, 𝑏, and 𝑐 are the lengths opposite them. Substituting in the values from this question, we have sin 𝐵 over eight is equal to sin of 36 degrees over five. Multiplying both sides of our equation by eight, we have sin 𝐵 is equal to eight multiplied by sin of 36 degrees divided by five. This is equal to 0.9404 and so on.

Our next step is to take the inverse sine of both sides of our equation. And 𝐵 is therefore equal to 70.1283 and so on. This is the measure of angle 𝐵 in degrees. However, we’re asked to give our answer to the nearest second. One way to convert this would be to press the degrees, minutes, and seconds button on our calculator. This gives us 70 degrees, seven minutes, and 42.04 seconds, which rounds to 70 degrees, seven minutes, and 42 seconds.

Alternatively, we could multiply the decimal part of our answer by 60, as there are 60 minutes in one degree. This gives us an answer of 70 degrees and 7.7007 and so on minutes. Once again, we multiply the decimal part by 60, as there are 60 seconds in a minute. Once again, this gives us an answer of 70 degrees, seven minutes, and 42.04 and so on seconds, and rounded to the nearest second, 70 degrees, seven minutes, and 42 seconds.

This value corresponds to the angle in the orange triangle. Recalling that the sin of 180 degrees minus 𝜃 is equal to sin 𝜃, we can find the second possible value of angle 𝐵 by subtracting 70 degrees, seven minutes, and 42 seconds from 180 degrees. This is equal to 109 degrees, 52 minutes, and 18 seconds. Since this angle is obtuse, it corresponds to angle 𝐵 in the blue triangle we sketched. We can therefore conclude that the two possible values of angle 𝐵 to the nearest second are 70 degrees, seven minutes, and 42 seconds and 109 degrees, 52 minutes, and 18 seconds.