Lesson Video: Right Triangle Trigonometry: Solving for a Side

Video Transcript

In this video, we will learn how to find the value of a missing side length in a right triangle by choosing the appropriate trigonometric ratio for a given angle. So what are these trig ratios? When we have a right triangle, we use the acronym SOHCAHTOA to help us remember the definitions of the trig ratios, sine, cosine, and tangent. We say that the sin of 𝜃 is equal to the opposite over the hypotenuse. The cos of angle 𝜃 is equal to the adjacent side length over the hypotenuse. And the tan of angle 𝜃 is equal to the opposite side length over the adjacent side length.

But in order to get these ratios correct, we need to label the triangle correctly. And that means we’ll always consider which angle we’re using. Here, this is our angle 𝜃. The opposite side is the side directly opposite the angle concerned. The adjacent side is the side next to the angle that is not the hypotenuse. And the hypotenuse is always the longest side of the right triangle. And it’s directly opposite the right angle. When we’re able to remember the trig ratios and correctly label our right triangle, we’re ready to start looking at how to calculate unknown lengths of the right triangle.

Here’s an example where we need to find a missing side length.

Find 𝑥 in a given figure. Give your answer to two decimal places.

The first thing we notice is that this is a right triangle. We know an angle and a side length. And that means to solve the problem, we’ll need to use the trig ratios. We remember with 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 adjacent. Our starting point is always the angle concerned. We’re given the angle 68 degrees. And so we can label the opposite side side 𝑥, the adjacent side, the side that is 11, and the hypotenuse is always opposite the right angle.

Once we’ve labelled these sides, we see that we’re given the side length of the adjacent side. And we’re interested in the side length of the opposite side. Since we’re dealing with the opposite and the adjacent, we’ll be looking at the tangent ratio. Since the tan of 𝜃 equals the opposite over the adjacent, we plug in 68 degrees for the angle. The opposite side is the side we’re trying to find, 𝑥, and the adjacent side measures 11.

In order to solve for 𝑥, we need to isolate it to get it by itself. We can do that by multiplying both sides of this equation by 11. 11 times tan of 68 degrees will equal the side length 𝑥. From here, to solve it, we’ll need to use a calculator. We’ll enter 11 times tan of 68. And it will give us 27.22595. If your calculator does not return this solution, you should check and make sure that it’s set to degree mode and not to radians.

We’ll take what we found in our calculator and plug it into our equation. The missing side of 𝑥 is 27.22595. We want it correct to two decimal places. To round to the second decimal place, we look to the right. Since there’s a five in the third decimal place, we need to round up. And we’ll get that 𝑥 equals 27 and twenty three hundredths, 27.23. As we aren’t given any units, it’s fine to leave it in this format. 𝑥 equals 27.23.

Here’s another example. This time, we’re missing two of the side lengths. And we need to solve for both of the missing sides.

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.

Notice how in these problems we’ve been dealing with missing side lengths as the numerator of the fraction in the ratio.

Let’s look at an example where we have a side length that ends up in the denominator of this ratio.

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.

Now let’s consider a word problem, a problem that doesn’t immediately seem to have anything to do with a triangle.

A person is trying to estimate the height of the Eiffel Tower. He measured a distance of 250 meters from the base of the tower. From that point, he measured the angle of elevation to the top of the tower to be 52 degrees. Use these measurements to approximate the height of the tower to the nearest meter.

When presented with a problem like this, the first thing that we should do is sketch out the information that we were given. We have an Eiffel Tower, and someone has measured 250 meters from the base of the tower. From that point, he measured an angle of elevation to the top of the tower to be 52 degrees. Once we have all of this information down into a diagram form, we should be able to see a right triangle forming.

The height of the tower forms a right angle with the base. The height is our unknown value and what we’re trying to solve for. So now we need to start at our angle of elevation and label the sides of the right triangle. The height is opposite to the angle we know. The 250-meter base is adjacent to the angle we know. And the other line from the person to the top of the tower is the hypotenuse.

In this problem, we’re not interested in the value of the hypotenuse. We’re dealing with the opposite side and the adjacent side, which means we’ll consider the ratios 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 adjacent. Based on the information we’re given, we need the tangent ratio.

If the tan of 𝜃 equals the opposite over adjacent, we can say the tan of 52 degrees equals ℎ, the height of the tower, over 250 meters. To get an estimate for ℎ, we’ll need to solve for ℎ to get ℎ by itself. And so we multiply both sides of the equation by 250. And then, we’ll see that 250 times tan of 52 degrees equals ℎ. When we plug that into a calculator, we get 319.9854 continuing. If we want to round to the nearest meter, we’ll be rounding to the nearest whole number. So we’ll look to the first decimal place and see that we should round up. The units we’re measuring in is meters. And so we would say that an estimate for the height of the Eiffel Tower based on the given information is 320 meters.

Let’s look at one final example where we’re not given a diagram.

Find the length of line segment 𝐴𝐶 given 𝐴𝐵𝐶 is a right-angled triangle at 𝐵, where sin of 𝐶 equals nine over 16 and 𝐴𝐵 equals 18 centimeters.

In this case, the first step should be to sketch a right triangle that meets these conditions. First, we have a right triangle. The right angle is at 𝐵. So we label our right angle 𝐵. And then we add 𝐴 and 𝐶. We’re told that 𝐴𝐵 measures 18 centimeters. And then we have this other piece of information that sin of 𝐶 equals nine over 16. This tells us that the angle we’re talking about is angle 𝐶. And if we think of our trig ratio acronym SOHCAHTOA, we know that the sin of an angle is equal to the opposite over the hypotenuse. And so if the angle we’re considering is 𝐶, the opposite side length will be the side 𝐴𝐵. And the hypotenuse is always the side opposite the right angle. And so that relationship is 9 to 16.

The key thing to remember here is that these relationships are ratios. And so sin of angle 𝐶 tells us that, for every nine units on the opposite side length, there will be 16 units on the hypotenuse side length. So we can say that if there are 18 centimeters as the opposite side, we know that nine times two equals 18. And when dealing with ratios or fractions, if we multiply by two in the numerator, we need to multiply by two in the denominator. 16 times two is 32. And so we could say that the hypotenuse must measure 32 centimeters. Line segment 𝐴𝐶 is the hypotenuse. And that measures 32 centimeters.

When we have right triangles and we need to solve for one of the sides, we have to remember our three trigonometric ratios and then follow these steps. One, label the sides in the triangle as opposite, adjacent, and hypotenuse relative to the known angle. Two, choose the correct trigonometric ratio which links the known side to the unknown side. And finally, substitute in the values and then solve.