Video: Using Trigonometric Ratios to Find Two Missing Lengths of a Right-Angled Triangle

Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.

03:07

Video Transcript

Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.

We notice that this is a right triangle. We’re given an angle and a side length, which means we can use trig ratios to solve for the two missing sides, remembering the acronym SOHCAHTOA. Sin of 𝜃 equals the opposite over the hypotenuse. Cos of 𝜃 equals the adjacent over the hypotenuse. And tan of 𝜃 equals the opposite over the hypotenuse [adjacent].

The key here is for us to label this triangle correctly. And to do that, we’ll use the given angle as our starting point. We label the side lengths relative to our given angle. The 𝑦 is the opposite side to the 40-degree angle. The 𝑥 is the adjacent side to the 40-degree angle. And the hypotenuse is always the side opposite the right angle.

First, let’s try to solve for 𝑦. If we’re solving for 𝑦 and we know the hypotenuse, we’ll use the sine ratio because the sin of 𝜃 is the opposite over the hypotenuse. And that means we can say that the sin of 40 degrees is equal to 𝑦 over 14. Since our goal is to solve for 𝑦, we’ll multiply both sides by 14. And then, we’ll see that 14 times sin of 40 degrees equals 𝑦.

When we plug that into our calculator, we get 8.99902 continuing. If you don’t get this answer in your calculator, then you should check and make sure that you’re operating in degrees and not in radians. We want our answer to three decimal places. So we look to the fourth decimal place where there is a zero, which means we’ll round down. 𝑦 is equal to 8.999. And the units we’re measuring are centimeters. So we say that 𝑦 equals 8.999 centimeters.

Next, we need to solve for 𝑥. And we can solve for 𝑥 with two different ratios. We could use the adjacent side and the hypotenuse side, which would be the cosine ratio. Or we could take what we found for 𝑦 and use that as the opposite side. And that would mean we would use the tangent ratio because we would have the opposite and adjacent sides. In this case, let’s use the hypotenuse as it will save us a little bit of writing.

We’re dealing with the cosine ratio. And that means we’ll have 𝑥 over 14. We’ll multiply both sides by 14. 14 times cos of 40 degrees will equal 𝑥. So 𝑥 will be equal to 10.72462 continuing. Rounded to the third decimal place means we need to round up to 10.725. Again, the units here will be measured in centimeters. And so we can say that 𝑥 is equal to 10.725 centimeters.