Question Video: Using the Law of Sines to Calculate an Unknown Length Mathematics

Video Transcript

For the given figure, 𝐵𝐶 is 11, the measure of angle 𝐴𝐶𝐵 is 44 degrees, and the measure of angle 𝐵𝐴𝐶 is 100 degrees. Work out the length of 𝐴𝐵. Give your answer to two decimal places.

Let’s move the information from the text of the question to the diagram. We are told that 𝐵𝐶 is 11, the measure of angle 𝐴𝐶𝐵 is 44 degrees, and the measure of angle 𝐵𝐴𝐶 is 100 degrees. We are required to find the length of 𝐴𝐵 which I’m calling 𝑐. We can solve this question using the sine rule which says that for any triangle 𝐴𝐵𝐶 — where the length of the side opposite vertex 𝐴 is lowercase 𝑎, the length of the side opposite vertex 𝐵 is lower case 𝑏, and the length of the side opposite vertex 𝐶 is lowercase 𝑐 — we have that the sine of the measure of the angle at vertex 𝐴, or just sine of uppercase 𝐴 for short, divided by lowercase 𝑎 is equal to the sine of uppercase 𝐵 divided by lowercase 𝑏 is equal to the sin of uppercase 𝐶 divided by lowercase 𝑐.

So for all three vertices, the sine of the measure of the angle and that vertex divided by the length of the side opposite that vertex is the same. And the same is true for the reciprocal 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵 is equal to 𝑐 over sin 𝐶. You can obtain this version of the sine rule simply by rearranging the previous version.

So what information were we given in the question? Well, we’re given the value of big 𝐴 or uppercase 𝐴 and little 𝑎, the lowercase 𝑎. So that’s the measure of the angle at the vertex 𝐴 as well as the length of the side opposite vertex 𝐴. We’re given the value of big 𝐶 and we are required to find the value of little 𝑐. So the form of the sine rule that we want is 𝑐 over sin 𝐶 is equal to 𝑎 over sin 𝐴.

We substitute in the values that we have from our diagram. We know that the value of big 𝐶 is 44 degrees, the value of little 𝑎 is 11, and the value of big 𝐴 is 100 degrees. Multiplying both sides by sin 44 degrees, we get that 𝑐 is equal to sin 44 degrees times 11 over sin 100 degrees. And putting this into our calculators, making sure first that we are in degree mode, we get 𝑐 is equal to 7.759 dot dot dot, the decimal expansion continues.

And so to two decimal places as required, the value of 𝑐, which was of course the length of side 𝑎𝑏, is 7.76.