Solving Right Triangles: Videos & Practice Problems

To solve any right triangle with one side and one angle, first find the missing angle by subtracting the known angle from 90 degrees. Use the sine function, where $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$, to find the missing side. For two sides given, apply the Pythagorean theorem $a^2+b^2=c^2$ to find the hypotenuse, then use trigonometric functions to find the angles.

Finding Missing Side Lengths

Hey, everyone, and welcome back. So up to this point, we've spent a lot of time talking about trig functions and right triangles, as well as some of the special case right triangles that allow us to solve problems super fast. Well, in this video, we're going to be learning about how we can find the missing side lengths for any kind of right triangle that we have, as long as we are given one side and one angle. Now this is a very important skill to have because you're not always going to have these special cases where you can use shortcuts. You're going to need to know how we can use the trigonometric functions and equations we've already learned about to solve any kind of right triangle.

So without further ado, let's get right into this. Now what we're going to do is take a look at this example, where we're asked to find all side lengths of the given triangle. Now our first step should be to find any missing angles in this triangle, because this gives us some nice options when we use our trigonometric functions. Now looking at the right triangle that we're given, I see that we have one side, which is the hypotenuse, and then we have this angle, which is 37 degrees, and nothing else is given to us. Now if we want to find this other non-right angle, what we can do is take the angle that we have here, and we can subtract it from 90 degrees.

So if I take 90 degrees minus our angle a, which we'll say that our angle a is 37 degrees, we'll have 90-37=53. So that means that our missing non-right angle that we have there is going to be 53 degrees, and that's our first step. Now our next step is going to be to choose a trigonometric function that includes one of the missing sides and the given side. Now what I'm going to do is see if I can find this missing side of the triangle, and this missing side is opposite to our 37-degree angle. So I'm going to use this angle to see if I can find this missing side.

Now the way that I can do this is by using the SOHCAHTOA memory tool because recall that SOHCAHTOA tells us how the trigonometric functions relate to the sides of the right triangle. Now because we have the hypotenuse or the long side of the triangle, I can use either the sine or the cosine to find this missing side. But what I'm going to do is use the sine, because we're going to have that the sine of our angle theta is equal to the opposite divided by the hypotenuse. And since I'm trying to find this missing side of the triangle, I'm going to use the angle that is opposite to that side. So we're going to have that the sine of our 37-degree angle is equal to the opposite side of this triangle, and the side opposite is the missing side x divided by the hypotenuse, which in this case is five.

So now that we have this equation set up, our third step is going to be to solve for x, which is the second side length of the triangle. What I'm first going to do is multiply both sides of this equation by five, and that's going to get the fives to cancel on the right side. And notice how that leaves us with just x by itself. And while the x is equal to 5×sin⁡(37°). So all you need to do is take five times the sine of 37 and plug it into your calculator, and make sure your calculator is in degree mode.

If you do this, you should get an approximate answer of three. Now on your calculator, it's gonna read as like 3.009075 and say I keep going on, but we can say that's about equal to three. So that means that our missing side x is three. Now our last step is going to be to find the final missing side using the Pythagorean theorem. And recall that this is what the Pythagorean theorem looks like.

So the way that I can do this is by recognizing that we have two of the sides, and we're missing this third side, which I'm going to call b. I'm going to say that this side we just calculated is a, and then the hypotenuse is always equal to c. So we have that a2+b2=c2, so we can say that three squared plus b squared is equal to c or five squared. So three squared is equal to nine, and then we have plus b squared, and that's equal to five squared, which is 25. Now from here what I can do is take nine and subtract it on both sides of this equation.

This is going to get the nines to cancel on the left side, leaving us with just b squared. And the last b squared is equal to 25 minus nine, which is 16. Now our last step is going to be to take the square root on both sides of this equation, canceling the square on the b, leaving us with just b. And while b is equal to the square root of 16, which is four. So that means that our missing side b is four, and notice how we were able to use trigonometric functions and the Pythagorean theorem to find all missing sides of this right triangle.

So this is the strategy you can use to solve any kind of right triangle, as long as you are given one side and one angle. So I hope you found this video helpful. Thanks for watching.

A right triangle with an angle of $31°$ has a hypotenuse of $10$. Calculate the side of the triangle opposite to the $31°$ angle (y), and the side adjacent to the $31°$ angle (x). Round your answer to 3 decimal places.

• A. $x=5.150,y=8.572$
• B. $x=8.572,y=5.150$
• C. $y=5.000,x=8.660$
• D. $y=8.660,x=5.000$

Answer: (B)

Finding Missing Side Lengths Example 1

Welcome back, everyone. So let's try this example. In this example, we are told the Grand Lighthouse on a coastal cliff stands 288 meters tall and is positioned approximately 2.3 kilometers inland from the shore of the sea. A seafarer on a sailboat directly in front of the lighthouse observes the top of the structure and records the angle of elevation as 3.4 degrees. Determine the distance in kilometers of the sailboat from the coastline to two decimal places.

Okay. So this is a story problem. And what we're going to do is see if we can use our understanding of right triangles and trigonometry to solve this problem. I drew a diagram of this situation. And I'll say, nothing's specifically drawn to scale here, but this is the situation, basically.

We have this boat, which is measuring to the top of this lighthouse. And we're told that the lighthouse is 288 meters tall. So that is going to be the height of this lighthouse. Now what we also have in this problem is this sailboat. And this sailboat over here is measuring a distance from sailboat to the top of the tower.

It says that this angle of elevation is 3.4 degrees. Now what we also have is that this lighthouse is approximately 2.3 kilometers inland from the shore of the sea. So this is where the lighthouse is, and then the shore of the sea is right about over there. And then this distance is going to be 2.3 kilometers. And keep in mind, there's a thousand meters in a single kilometer.

So 2.3 kilometers is the same thing as 2,300 meters. This is the distance there. And what we're trying to do is actually find this distance, which is the distance from the shore to the boat. So we'll call this distance d. Now to go about solving this problem, what we need to do is just think about how we can relate all of these sides.

If you look closely, you can see that we actually have a right triangle, because this is a right triangle that we're looking at. So I can use SOHCAHTOA. I notice that the hypotenuse is not a value that is given to us. So using the sine and the cosine might not be the best idea. But I notice that we have the opposite side of this triangle as well as the adjacent side, so the tangent is our best way to go.

We have that the tangent of our angle theta is going to equal the opposite side of this triangle divided by the adjacent side. And, from this triangle, if we go opposite the angle that we have, it's 288 meters. So we're going to have that the tangent is equal to 288 meters divided by, and then we have the adjacent side, which is this whole side of the triangle. So it's going to be 2,300 plus our distance d. And then this tangent of our angle is 3.4 degrees.

tan ( 3.4 � ) = 288 2300 + d

So this is going to be the angle within the tangent operation. From here, all I need to do is use some algebra to isolate this distance. I'm going to multiply both sides of this equation by 2,300 plus d.

This will cancel the numbers on the right side of the equation, so all we're going to be left with is 288. We're going to have 2,300 plus d times the tangent of 3.4 degrees. I can distribute this tangent into these parentheses.

We're going to have 2,300 times the tangent of 3.4 degrees, which is 136.65. This is going to be plus d times the tangent of 3.4 degrees. All this is going to be equal to 288. Subtracting 136.65 on both sides, we get 151.35, which is all going to be equal to d times the tangent of 3.4 degrees.

Now, all I need to do is get d by itself by dividing both sides of this equation by the tangent of 3.4 degrees. Doing this will yield a value of approximately 2,547.6 meters. We are asked for the answer in kilometers, so we need to convert this to kilometers by dividing by a thousand. So our distance is going to be about 2.55 kilometers. And this is the distance from the sailboat to the shore. So 2.55 kilometers is the distance, and that is the answer to this problem.

Hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.

Finding Missing Angles

Hey, everyone. So up to this point, we have worked a lot with right triangles. We've discussed some special case right triangles, as well as how we can solve any kind of right triangle that has one angle and one side and nothing else. Well, in this video, we're going to be learning about some strategies to solve any kind of right triangle that has two or more sides given to you. So if you have two of the side lengths or more, you can actually solve for all other sides and angles in the right triangle.

And it's very important to have this skill because you're going to see these types of problems show up a lot through this course. So without further ado, let's see how we can solve these types of problems. Now, here we have an example where we're asked to find the angles of the given triangle, and we'll start with situation a. In this situation, notice that we have two sides and we have two missing angles. This is the 90-degree right angle right there, and we need to find the missing angles.

Now our first step is going to be, if there are any missing sides, to use the Pythagorean theorem to find that side. And I can see that we are missing the hypotenuse, and recall that the Pythagorean theorem looks something like this: a 2 + b 2 = c 2 . Now a and b are going to be the two sides of the triangle that are not the hypotenuse, so I'll say that this is a and that's b. These are interchangeable.

They could be either way. And then c always has to be the hypotenuse or the longest side. So using this equation, we're going to have 12 squared plus five squared is equal to c squared: 12 2 + 5 2 = c 2 .

Now 144 plus 25 comes out to 169. And if I go ahead and solve for c, I can take the square root on both sides of this equation, leaving me with c is equal to the square root of 169, which is 13. So that means that 13 is the missing side of this right triangle. So now that we've found this missing side, our next step is going to be to for any of the non-90-degree angles, to write a trig equation that involves the known sides.

And to find a trig equation that would do this, we can use SOHCAHTOA, which is the memory tool for each trig function relating to the sides of the triangle. Now what I'm gonna do is take a look at angle x as our reference here, and I'm going to use the sine, which is opposite over hypotenuse. So we'll have that the sine of our angle x is equal to the opposite side of the triangle, which in this case would be 12 divided by the hypotenuse, which is the long side or 13: sin ( x ) = 12 13 .

Now to go ahead and find x, which this is what we're setting up in step two, what we need to do is take the inverse on both sides of the equation for the trig function to find our angle. So I'll take the inverse sine on the left side and the inverse sine on the right side: x = sin −1 ( 12 13 ) .

Now from here, what you can do is take this value and plug it into a calculator. And if you plug this into a calculator, you should get an approximate value of 67.38 degrees. Now to simplify this, I'm going to round this to just 67. So we're going to say that this missing angle x is 67 degrees.

So now that we've found this angle x, our final step is going to be to find this angle y. And we can do that by just recognizing that these two angles are going to be complementary, meaning that we can just take 90 degrees and subtract off the angle we just calculated. So what I can do is say that our angle y is going to be equal to 90 degrees minus this 67-degree angle: y = 90 − 67 . And 90 minus 67 turns out to be 23. So that means the missing angle y is 23 degrees.

And notice how we were able to use the SOHCAHTOA memory tool as well as this relationship with the Pythagorean theorem to solve for all of the missing angles in the right triangle when all we were given were two of the sides. Now to make sure we're understanding this, let's go ahead and try one more example to really make sure we've got this down. So in this example, we're already given all the sides of the right triangle, so we don't really need to do this step where we use the Pythagorean theorem. So what I'm going to do is this next step where we need to find a trig equation that represents one of the angles. Now in this example, I'll focus on angle b here, and this time I'm going to use the cosine to find this missing angle.

Since we have all the sides, you could actually use any one of these trig functions you wanted to, but just for the sake of practice, I'm gonna use the cosine. So we're gonna have that the cosine of our angle B is equal to adjacent over hypotenuse: cos ( B ) = adjacent hypotenuse : the adjacent side of this triangle is 8, and then the hypotenuse is the long side or 17:

Now what I can do from here is take the inverse cosine on both sides of the equation to get the angle b by itself: B = cos −1 ( 8 17 ) .

Now what you can do is plug this value into your calculator, and you should get an approximate value of 61.92 degrees. But to simplify this, I'm going to round this to 62 degrees. So we'll say that angle b is 62 degrees, and then all we need to do is find angle a. And we can do that by taking 90 degrees and subtracting it from the angle that we just calculated: a = 90 − 62 .

And 90 degrees minus 62 degrees comes out to 28 degrees, meaning 28 degrees is our missing angle. So that is how you can solve for all of the angles if you're only given two or more of the sides of a right triangle. So I hope you found this video helpful. Thanks for watching, and let me know if you got any questions.

Given the right triangle below, calculate all missing angles in degrees (round your answer to 3 decimal places.

• A. $x=60.000°,y=30.000°$
• B. $x=26.565°,y=63.435°$
• C. $=63.435°,y=26.565°$
• D. $x=30.000°,y=60.000°$

Answer: (B)

Finding Missing Angles Example 2

Welcome back, everyone. So here we have an example where we need to use our understanding of right triangles and trigonometry to see if we can solve this story problem. Let's see what we've got. Here we have a hiking path traced from a mountain lodge at an elevation of 6,500 feet to a scenic viewpoint in a canyon at an elevation of 4,300 feet. The path spans 4,400 feet, and we're asked to determine the angle of inclination of the hiking path.

Okay. Now to understand this, I actually drew this sketch out just to give us a general understanding of what's going on. And I will say that the sizes here are not necessarily drawn to scale, but this is going to give us a general understanding of where things are positioned. Now we have this mountain lodge at an elevation of 6,500 feet, which is going to be up here. So this distance would be 6,500 feet.

Now we have that we're trying to reach a destination that is at an elevation of 4,300 feet, and that is going to be this distance down here. We're trying to see what this is going to be, because the path spans 4,400 feet. This means that this path to get to our destination is 4,400 feet. We're trying to find the angle of inclination, and we can find this if we draw a straight line right here.

Because this straight line is going to show us what the angle of inclination is, which is that angle. Notice that we have a right triangle that forms when we do this. But the height of this side of the triangle is not going to be the whole 6,500 feet. It's going to be some other height that we'll call h. The reason is that 6,500 feet is this entire distance, not just that distance.

So what we need to do is figure out how we can use all this information to solve for this angle. Well, something that I see is we can actually figure out what this height is. Notice that we have this distance as well as that distance. Our height h is going to be 6,500 feet minus 4,300 feet. That's going to give us this distance from 4,300 feet of elevation to the mountain lodge.

So 6,500 minus 4,300 will give us 2,200 feet. This right here is the height that we are dealing with. Now that we have the height, what we need to do is solve for the angle. We can solve for this angle using SOHCAHTOA, a memory tool that we use to relate the trigonometric functions to the sides of the right triangle.

Now what I noticed in this problem is that we have the opposite side of this triangle. So if we're looking at this angle, the opposite side is going to be 2,200 feet, and the hypotenuse is 4,400 feet. The trigonometric function that uses the opposite and the hypotenuse is the sine. So we can see that the sine of our angle theta is equal to the opposite side of this triangle, which is 2,200 feet divided by the hypotenuse, which is 4,400 feet. Now to solve for our angle theta, I'm going to take the inverse sine on both sides of the equation.

That's when we get the sine and the inverse sine to cancel on the left side, leaving us with just theta. And theta is going to equal the inverse sine of 2,200 over 4,400. Now 2,200 over 4,400 is something that can reduce. This is equivalent to the inverse sine of one-half, because 2,200 goes into 4,400 two times, so this would reduce to a half. And the inverse sine of one half comes out to an angle of 30 degrees.

So 30 degrees is our angle of inclination, and the solution to this problem. So that's how you can solve this. Hope you found this video helpful. Thanks for watching.

Here’s what students ask on this topic:

How do you find the missing side of a right triangle using trigonometric functions?

To find the missing side of a right triangle using trigonometric functions, you can use the SOHCAHTOA memory tool. First, identify the given side and angle. For example, if you know the hypotenuse and an angle, you can use the sine function: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Rearrange the equation to solve for the missing side: $\mathrm{opposite}=\mathrm{hypotenuse}\times \sin \left(\theta \right)$. Plug the values into your calculator, ensuring it is in degree mode, to find the missing side length.

How do you calculate the missing angle in a right triangle?

To calculate the missing angle in a right triangle, use the complementary angle property or trigonometric functions. If one angle is given, subtract it from 90°: $\mathrm{missing}\mathrm{angle}=90-\mathrm{given}\mathrm{angle}$. If two sides are given, use inverse trigonometric functions. For example, if you know the opposite and hypotenuse, use the sine function: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Take the inverse sine: $\theta =\sin ⁻¹\left(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\right)$. Use a calculator to find the angle.

What is the Pythagorean theorem and how is it used to solve right triangles?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: $a^2+b^2=c^2$. To find a missing side, rearrange the equation. For example, if the hypotenuse is given, solve for a leg: $b^2=c^2-a^2$. Take the square root: $b=\sqrt{c^2-a^2}$. This theorem is essential for solving right triangles when two sides are known.

How do you use SOHCAHTOA to solve for missing sides and angles in a right triangle?

SOHCAHTOA is a mnemonic for trigonometric functions: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent). To solve for a missing side, choose the function based on the given side and angle. For example, if the hypotenuse and an angle are given, use sine: $\sin \left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. Rearrange to find the opposite side: $\mathrm{opposite}=\mathrm{hypotenuse}\times \sin \left(\theta \right)$. To find an angle, use the inverse function. For example, if opposite and hypotenuse are known, use: $\theta =\sin ⁻¹\left(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\right)$. Plug values into a calculator to solve.

What are complementary angles in a right triangle, and how are they calculated?

Complementary angles in a right triangle are the two non-right angles that sum to 90°. If one angle is known, the other can be calculated by subtracting the known angle from 90°: $\mathrm{missing}\mathrm{angle}=90-\mathrm{given}\mathrm{angle}$. For example, if one angle is 37°, the other is $90-37=53$. This property simplifies calculations when solving right triangles.