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

Video Transcript

For the given figure, 𝐴𝐵 is equal to three and 𝐵𝐶 is equal to 𝑎. Use the law of sines to work out 𝑎. Give your answer to two decimal places.

Let’s begin by adding the given measurements to our triangle. We can see that we have a nonright-angled triangle for which we know the measure of two angles and the length of one side. To find the length of the side labelled 𝑎, we’ll need to use the law of sines: 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶.

Alternatively, that can be written as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. We only need to use one of these forms. Since we’re trying to calculate the length of one of the sides, we’ll use the first form. It doesn’t particularly matter either way. But by using the first form here, it will minimize the amount of rearranging we’ll need to do to solve the equation.

Next, we’ll label the sides of the triangle. The side opposite the angle 𝐴 is already given by the lowercase 𝑎. The side opposite the angle 𝐵 is lowercase 𝑏, and the side opposite the angle 𝐶 is lowercase 𝑐. For the law of sines, we usually only need to use two parts. We don’t know the side labelled lowercase 𝑏, so we’re going to use 𝑎 over sin 𝐴 and 𝑐 over sin 𝐶.

Let’s substitute what we know into this formula. That gives us 𝑎 over sin 64 is equal to three over sin 31. To solve this equation and work out the value of 𝑎, we’ll need to multiply both sides by sine of 64. 𝑎 is therefore equal to three over sine of 31 multiplied by sine of 64.

If we put that into our calculator, we get that 𝑎 is equal to 5.2353 and so on. Correct to two decimal places, 𝑎 is equal to 5.24. Notice how there were no units provided in the question, so no units are required in our answer.