Solving Right Triangles - Online Tutor, Practice Problems & Exam Prep

To solve right triangles, use trigonometric functions and the Pythagorean theorem. For one side and one angle, find missing angles by subtracting from 90 degrees. Use SOHCAHTOA to relate sides and angles, such as $sin\left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$. For two sides, apply the Pythagorean theorem $a²+b²=c²$ to find the hypotenuse, then use trigonometric ratios to find 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 these 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. 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 minus 37, which is 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 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. 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 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 5.

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 5, and that's going to get the fives to cancel on the right side. Notice how that leaves us with just x by itself. And we'll have that x is equal to 5 times the sine of 37 degrees. So all you need to do is take 5 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 3. Now, on your calculator, it's going to read as, like, 3.009075, and it's going to keep going on. But we can say that that's about equal to 3. So that means that our missing side x is 3.

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 2 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 a squared plus b squared equals c squared. So we can say that 3 squared plus b squared is equal to c or 5 squared. So 3 squared is equal to 9, and then we have plus b squared, and that's equal to 5 squared, which is 25.

Now from here what I can do is take 9 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 b². And the last b² is equal to 25 minus 9, 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 we'll have that b is equal to the square root of 16, which is 4. So that means that our missing side b is 4. 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. 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. $y=8.660,x=5.000$

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 2 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. So what I did is drew a diagram of this situation, and I'll say nothing's, like, specifically drawn to scale here, but this is the situation, basically. And what we have is 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 we have this sailboat, and the sailboat over here is measuring a distance from sailboat to the top of the tower. And 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 are 1000 meters in a single kilometer. So 2.3 kilometers is the same thing as 2300 meters. So 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. And if you look closely, you can see that we actually have a right triangle which forms because this is a right triangle that we're looking at, so I can use SOHCAHTOA.

Now what I notice is 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 noticed that we have the opposite side of this triangle, as well as the adjacent side, so the tangent is our best way to go. So 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 what I can see from this triangle is that if we go opposite to 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 2300 plus our distance \( d \), and then this tangent of our angle is 3.4 degrees. So this is going to be the angle within the tangent operation. And now from here, all I need to do is use some algebra to get this distance by itself, and I'm going to go ahead and do this over here.

So what I'm first going to do is take both sides of this equation. I'm going to multiply it by 2300 plus \( d \). So I'm going to apply the left and the right side by this quantity, and what that is going to allow me to do is cancel these numbers on the right side of the equation. So all we're going to be left with is 288. And then on the left side, we're going to have 2300 plus \( d \) times the tangent of 3.4 degrees. What I can actually do is take this tangent, and I can distribute it into these parentheses. We're going to have 2300 times the tangent of 3.4 degrees, and this is going to be plus \( d \) times the tangent of 3.4 degrees. Now what I'm going to do from here is 2300 times the tangent of 3.4. I'm going to put that into my calculator to simplify this and make this equation a little bit easier.

So putting this into a calculator, this number comes out to about 136.65. This is going to be plus \( d \) times the tangent of 3.4, and this is all going to be equal to 288. Now what I can do at this point is take this 136.65 and subtract it on both sides of the equation. That is going to get these to cancel on the left side. And then on the right side, we'll have 288 minus this number. Subtracting these 2 will give you 151.35, and that is all going to be equal to \( d \) times the tangent of 3.4. Now at this point, all I need to do is get \( d \) by itself, and I can do that by dividing both sides of this equation by the tangent of 3.4. And doing this will allow me to cancel the tangent of 3.4 on the left side of the equation, leaving us with just \( d \). And then on the right side of the equation, we'll have 151.35 divided by the tangent of 3.4. Doing this should give you a value of approximately 2547.6 meters. And keep in mind that we are asked for the answer in kilometers in this problem. So what we need to do is convert this to kilometers by dividing this by 1000. So there's 1000 meters in a single kilometer. So our distance is going to be this number if we move the decimal place back 3 points, it's going to be about 2.55 kilometers. And this right here is the distance from the sailboat to the shore. So 2.55 kilometers is the distance, and that is the answer to this problem. So 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 2 or more sides given to you. So if you have 2 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 2 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:

a2 + b2 = c2

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:

122 + 52 = c2

Now 122 is 144, and 52 is 25. And that's all going to be equal to c2. 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 = 169

So that means that 13 is the missing side of this right triangle, so that's what we have for the hypotenuse. So now that we 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 going to 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. Now to go ahead and find x, which this is what we're setting up in step 2, 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. That leaving me with just our angle x is equal to the inverse sine of 12 over 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 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. 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 2 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 going to use the cosine. So we're going to have that the cosine of our angle b is equal to adjacent over hypotenuse. So 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. That's going to get this cosine to cancel, giving us that b is the inverse cosine of 8 over 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. So we're going to say that angle a is equal to 90 degrees minus what we just calculated for angle b, which is 62 degrees. 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 2 or more sides of a right triangle. So I hope you found this video helpful. Thanks for watching, and let me know if you have 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. $x=30.000°,y=60.000°$

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. So let's see what we've got. Here we have that a hiking path can be 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. We have that 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 that this mountain lodge is 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.

And what we're trying to do is see what this is going to be, because we have that the path spans 4,400 feet. So that means that this path to get to our destination is 4,400 feet. And what we're trying to do is 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. Now 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. And the reason why is because 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 because notice that we have this distance as well as that distance. And what our height h is going to be is going to be the 6,500 feet minus the 4,300 feet, because that's going to give us this distance from the 4,300 feet of elevation to the mountain lodge. So 6,500 minus 4,300 will give us 2,200 feet. And this right here is the height that we are dealing with.

So now that we have the height, what we need to do is solve for the angle, and we can solve for this angle using SOHCAHTOA. SOHCAHTOA is this 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 then the hypotenuse is 4,400 feet. And 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 / 4,400. Now 2,200 over 4,400 is something that can reduce. This is equivalent to the inverse sine of 1 half, because 2,200 goes into 4,400 two times, so this would reduce to a half. And the inverse sine of 1 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. For example, if you know one angle (θ) and the hypotenuse (c), you can find the opposite side (a) using the sine function: $\sin \left(\theta \right)=\frac{a}{c}$. Rearrange to solve for a: $a=c\cdot \sin \left(\theta \right)$. Similarly, you can use cosine for the adjacent side: $\cos \left(\theta \right)=\frac{b}{c}$, and rearrange to solve for b: $b=c\cdot \cos \left(\theta \right)$.

How do you find the missing angles in a right triangle?

To find the missing angles in a right triangle, start by using the fact that the sum of the angles in a triangle is 180 degrees. Since one angle is 90 degrees, the other two angles must sum to 90 degrees. If you know one of these angles (θ), the other angle (α) can be found by subtracting θ from 90 degrees: $\alpha =90°-\theta$. Additionally, you can use trigonometric functions to find angles if you know the sides. For example, if you know the opposite (a) and adjacent (b) sides, use the tangent function: $\tan \left(\theta \right)=\frac{a}{b}$, and then use the inverse tangent to find θ: $\theta =\tan ⁻¹\left(\frac{a}{b}\right)$.

How do you use the Pythagorean theorem to find the hypotenuse of a right triangle?

To find the hypotenuse of a right triangle using the Pythagorean theorem, use the formula: $c^2=a^2+b^2$, where c is the hypotenuse, and a and b are the other two sides. Rearrange to solve for c: $c=\sqrt{a^2+b^2}$. For example, if a = 3 and b = 4, then $c=\sqrt{3{²}^{}+4{²}^{}}=\sqrt{25}=5$.

What is SOHCAHTOA and how is it used in solving right triangles?

SOHCAHTOA is a mnemonic device to remember the relationships between the sides and angles of a right triangle using trigonometric functions. It stands for: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. To use SOHCAHTOA, identify the sides relative to the given angle (θ). For example, if you need to find the opposite side (a) and you know the hypotenuse (c) and angle (θ), use the sine function: $\sin \left(\theta \right)=\frac{a}{c}$. Rearrange to solve for a: $a=c\cdot \sin \left(\theta \right)$.

How do you solve a right triangle if you are given two sides?

To solve a right triangle when given two sides, first use the Pythagorean theorem to find the third side. For example, if you know sides a and b, find the hypotenuse c using: $c^2=a^2+b^2$. Then, use trigonometric functions to find the angles. For instance, to find angle θ, use the tangent function: $\tan \left(\theta \right)=\frac{a}{b}$, and then find θ using the inverse tangent: $\theta =\tan ⁻¹\left(\frac{a}{b}\right)$. Finally, find the remaining angle by subtracting θ from 90 degrees.