Video Transcript
𝐴𝐵𝐶 is a triangle where the measure of angle 𝐴 is 40 degrees, 𝑎 is equal to 17 centimeters, and 𝑏 is equal to 23 centimeters. If the triangle exists, find all the possible values for the other lengths and angles, giving the lengths to two decimal places and the angles to the nearest second.
We recall that when dealing with triangles, one way of calculating missing lengths and angles is using the sine rule or law of sines. This states that 𝑎 over the sin of angle 𝐴 is equal to 𝑏 over the sin of angle 𝐵 which is equal to 𝑐 over the sin of angle 𝐶 where lowercase 𝑎, 𝑏, and 𝑐 are the side lengths opposite the angles 𝐴, 𝐵, and 𝐶.
In this question, we are told that the measure of angle 𝐴 is 40 degrees, the side length 𝑎 is equal to 17 centimeters, and the side length 𝑏 is equal to 23 centimeters. We will begin by substituting these values into the formula to help us calculate the measure of angle 𝐵. This gives us 17 over the sin of 40 degrees is equal to 23 over the sin of angle 𝐵. Cross multiplying here, we get 17 multiplied by the sin of angle 𝐵 is equal to 23 multiplied by the sin of 40 degrees. We can then divide both sides of this equation by 17.
Finally, we take the inverse sine of both sides of this equation such that angle 𝐵 is equal to the inverse sin of 23 multiplied by the sin of 40 degrees all divided by 17. Typing the right-hand side into our calculator gives us 60.4184 and so on. This is the answer in degrees, and we need to give our angle to the nearest second. Using the degrees-minutes-seconds button on a scientific calculator, this can be converted to 60 degrees, 25 minutes, and six seconds. One possible measure of angle 𝐵 is 60 degrees, 25 minutes, and six seconds.
We know that the angles in a triangle sum to 180 degrees. This means that the measures of angle 𝐴, angle 𝐵, and angle 𝐶 must sum to 180. Substituting in our values for angle 𝐴 and 𝐵, we have 40 degrees plus 60 degrees, 25 minutes, and six seconds plus angle 𝐶 is equal to 180 degrees. We can add angles 𝐴 and 𝐵 and then subtract 100 degrees, 25 minutes, and six seconds from both sides. This gives us that angle 𝐶 is equal to 79 degrees, 34 minutes, and 54 seconds. When the measure of angle 𝐵 is 60 degrees, 25 minutes, and six seconds, the measure of angle 𝐶 is 79 degrees, 34 minutes, and 54 seconds.
Our next step is to use the sine rule or law of sines once again to calculate the side length 𝑐. Substituting in our values, we have 17 over the sin of 40 degrees is equal to 𝑐 divided by the sin of 79 degrees, 34 minutes, and 54 seconds. This means that 𝑐 is equal to 17 over the sin of 40 degrees multiplied by the sin of 79 degrees, 34 minutes, and 54 seconds. Typing the right-hand side into the calculator gives us 26.0112 and so on. To two decimal places, side length 𝑐 is equal to 26.01 centimeters.
At this stage, it appears as if we have completed the question as we have calculated the measure of the two missing angles and the missing side length. However, when dealing with problems of this type, we need to recall our CAST diagram. This tells us that there are two possible values of 𝜃 for which the value of sin 𝜃 is the same. As well as getting an answer of 60 degrees, 25 minutes, and six seconds for angle 𝐵, we could also subtract this value from 180 degrees. This gives us 119 degrees, 34 minutes, and 54 seconds. This means that an alternative value for the measure of angle 𝐵 is 119 degrees, 34 minutes, and 54 seconds.
Once again, the sum of our angles must equal 180 degrees. We know that angle 𝐴 is equal to 40 degrees, so 40 degrees plus 119 degrees, 34 minutes, and 54 seconds plus angle 𝐶 must equal 180 degrees. We can then subtract 159 degrees, 34 minutes, and 54 seconds from both sides of our equation to calculate the value of angle 𝐶. Angle 𝐶 is equal to 20 degrees, 25 minutes, and six seconds. This is the measure of this angle when angle 𝐵 is equal to 119 degrees, 34 minutes, and 54 seconds.
We can now use the sine rule once again to calculate the second possible value of side length 𝑐. Substituting in the values of angles 𝐴 and 𝐶 as well as side length 𝑎, we get 17 over the sin of 40 degrees is equal to 𝑐 over the sin of 20 degrees, 25 minutes, and six seconds. This gives us a value of 𝑐 equal to 9.2267 and so on. To two decimal places, side length 𝑐 is equal to 9.23 centimeters.
When the measure of angle 𝐴 is 40 degrees, side length 𝑎 is 17 centimeters, and side length 𝑏 is 23 centimeters, there are two possible values for angle 𝐵, angle 𝐶, and side length 𝑐. Firstly, we have 60 degrees, 25 minutes, and six seconds; 79 degrees, 34 minutes, and 54 seconds; along with 26.01 centimeters. Alternatively, we have 119 degrees, 34 minutes, and 54 seconds; 20 degrees, 25 minutes, and six seconds; and 9.23 centimeters.
