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.

03:54

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 trig ratios. And to remember those, we’ll use SOHCAHTOA. 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 next to that 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’ll need to divide both sides of the equation by sin of 47 degrees, like this. 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. This would mean you would solve for tan of 47 degrees is equal to 28 over 𝑥. Or if you used the hypotenuse and the adjacent side, you could say cos of 47 degrees equals 𝑥 over 38.285.

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 26.1104. We round to the third decimal place. And we get 𝑥 is equal to 26.110, being measured in centimeters, which means we found the two missing side lengths.

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