Question Video: Using Trigonometric Ratios to Find Two Missing Lengths of a Right-Angled Triangle Mathematics • 11th Grade

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

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Video Transcript

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

We have a right triangle. We’re given an angle and a side length and asked to find the two missing sides. To do this, we’ll need our trigonometric ratios. And to remember those, we’ll use SOH CAH TOA. The sin of 𝜃 equals the opposite over the hypotenuse. The cos of 𝜃 equals the adjacent over the hypotenuse. And the tan of 𝜃 equals the opposite over the adjacent. The key to solving these problems consistently is to correctly label the triangle. And we label them relative to the angle that we’re given. This is our starting point. The opposite side length is the side length directly opposite this angle. The adjacent side is between this angle and the right angle. And the hypotenuse is always opposite the right angle.

Once a triangle is labeled, we’re ready to identify which of the ratios we need. If we start by finding side length 𝑦, the hypotenuse, and we already know the opposite side, 28 centimeters, we need to use the sine ratio, as the sine ratio involves the opposite side length and the hypotenuse. The ratio would look like this. sin of 47 degrees is equal to 28 over 𝑦. When our variable is in the denominator, it will take two steps to find the value.

The first thing we would do is multiply both sides of the equation by 𝑦. When we do that, we get 𝑦 times sin of 47 degrees equals 28. If the goal is to isolate 𝑦, then at this point, we need to divide both sides of the equation by sin of 47 degrees. And then on the left we’ll just have 𝑦, and on the right we’ll have 28 over sin of 47 degrees.

When we plug that into the calculator, we get 38.28516 continuing. We need to round it to three decimal places. This value rounds down to 38.285. The sides are being measured in centimeters, so the units here would be centimeters. And that means we found one of the missing sides.

To find the side length 𝑥, we’ll have two choices. We could use the hypotenuse we just found, 38.285. If we did that, we’d be dealing with the adjacent side and the hypotenuse, which would be the cosine relationship. Or we could use the 28-centimeter side. In that case, we would be using the opposite side and the adjacent side and would need the tangent ratio.

In this case, let’s practice having the 𝑥-variable in the denominator. tan of 47 degrees equals 28 over 𝑥. To solve for 𝑥, we first multiply both sides of the equation by 𝑥. Then, we can say that 𝑥 times tan of 47 degrees equals 28. To isolate 𝑥, we divide both sides of the equation by tan of 47 degrees. And so we say that 𝑥 equals 28 over the tan of 47 degrees, which gives us 26.11042 continuing. We round to the third decimal place, and we get that 𝑥 is equal to 26.110. This is measured in centimeters. And so we found the two missing side lengths. To three decimal places, 𝑥 is equal to 26.110 centimeters and 𝑦 is equal to 38.285 centimeters.