Video: Using the Law of Sines to Calculate an Unknown Length in a Triangle

Video Transcript

𝑋𝑌𝑍 is a triangle, where 𝑌𝑍 is equal to eight centimeters, the measure of the angle at 𝑌 is 22 degrees, and the measure of the angle at 𝑍 is 23 degrees. 𝑊 lies on the line 𝑌𝑍, where the line 𝑋𝑊 is perpendicular to the line 𝑌𝑍. Find the length of the line 𝑋𝑊, giving your answer to two decimal places.

Let’s start by sketching this out. Remember, a sketch doesn’t need to be to scale, but it’s sensible to keep it roughly in proportion so we can check the suitability of any answers we get. Here we have a non-right-angled, triangle with the length of one side known. We also know two of the angles.

We can calculate the measure of the angle at 𝑋 by using the fact that angles in a triangle add to 180 degrees. Subtracting the measure of the angle at 𝑌 and the measure of the angle at 𝑍 from 180 gives us 180 minus 22 plus 23. The measure of the angle at 𝑋 is 135 degrees. Once we have the measure of the angle at 𝑋, we can use the law of sins to calculate either of the missing sides. We know to use this over the law of cosines since that requires at least two known sides.

The law of sins says that 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. That’s sometimes written as sin 𝐴 over 𝑎 equal sin 𝐵 over 𝑏, which equals sin 𝐶 over 𝑐. We can use either of these forms. However, since we’re trying to find the missing side, it’s sensible to use the first form to minimize the amount of rearranging we need to do. We would use the second form if we’re trying to calculate the measure of one of the angles.

We can change the letters in our formula to suit the problem. In that case, it’s 𝑥 over sin 𝑋, 𝑦 over sin 𝑌, and 𝑧 over sin 𝑍. We can also label the sides of the triangle as shown. The side opposite angle 𝑋 is lowercase 𝑥, the side opposite angle 𝑌 is lowercase 𝑦, and the side opposite 𝑍 is lowercase 𝑧.

At this stage of the problem, we can choose whether we want to calculate the length of the side marked 𝑦 or the side marked 𝑧. Let’s choose the side marked 𝑧. We’re going to use the formula 𝑥 over sin 𝑋 equals 𝑧 over sin 𝑍. Substituting all the values we know into the formula gives us eight over sin 135 equals 𝑧 over sin 23.

To solve this equation, we’ll multiply both sides by sin of 23. 𝑧 is equal to eight over sin of 135 multiplied by sin of 23, which is 4.42 centimeters. We won’t round this answer just yet. Instead, we’ll use its exact form in our next calculations.

We were told that the line 𝑋𝑊 is perpendicular to the line 𝑌𝑍, so we now have a right-angled triangle 𝑋𝑊𝑌. We can use right angle trigonometry to find the length of the line 𝑋𝑌. Let’s call that 𝑎. Labeling our triangle, we can see that the side 𝑋𝑊 is opposite to the angle. And we’ve calculated the length of the side 𝑋𝑌, which is now the hypotenuse of our triangle. This means we need to use the sine ratio to calculate the length that we’ve called 𝑎.

Substituting the values we know into this formula gives us sin of 22 is equal to 𝑎 over 4.42. And to solve this equation to calculate the value of 𝑎, we’ll multiply both sides by 4.42. We’ll try and use the exact value we calculated earlier. 𝑎 is equal to sin of 22 multiplied by 4.42, which is 1.6559 and so on. Correct to two decimal places, the length of 𝑋𝑊 is 1.66 centimeters.