Solving Right Triangles - Video Tutorials & Practice Problems

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 triangles. 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 will say that our angle A is 37 degrees, we 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 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 Sokoto, a memory tool because we're called it Sokoto 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 sign or the cosine to find this missing side. But what I'm going to do is use the sign because we're going to have that the sign 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 sign 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 five times the sign of 37 degrees, so all you need to do is take five times the sign 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 is like 3.009075. And they keep on, but we can say that 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 A squared plus B squared equals C squared. 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 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 the 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 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. So in this example, we are told the grand lighthouse on a coastal cliff stands 288 m 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. OK? 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 the 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 200 88 m 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 this sailboat over here is measuring a distance from the sailboat to the top of the tower. And it says that this angle of elevation is 3.4 degrees. Now, 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 gonna be 2.3 kilometers. And keep in mind there's 1000 m in a single kilometer. So 2.3 kilometers is the same thing as 2300 m. 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 Sokoto. Now, what I noticed is that the hypotenuse is not a value that is given to us. So using the sign 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 is 288 m. So we're going to have that the tangent is equal to 288 m 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 gonna 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 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. But what I can actually do is take this tangent and I can distribute it into these parentheses. So we're going to have 2300 times the tangent of 3.4 degrees and then is going to be plus D times the tangent of 3.4 degrees. Now, what I'm gonna do from here is 2300 times the tangent of 3.4. I'm gonna 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 100 and 36.65. This is gonna 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 1 36.65 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 will have 288 minus this number subtracting these two will give you 1 51.35. 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 with just D and then on the right side of the equation, we will have 1 51.35 divided by the tangent of 3.4. Doing this should give you a value of approximately 2547.6 m. 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 1000 m in a single kilometer. So our distance is going to be this number. If we move the decimal place back, three points is gonna be about 2.55 around this four up 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 I 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 the 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. So we have A squared plus B squared is equal to C squared. 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, that 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. Now 12 squared is 100 and 44 and five squared is 25. And that's all going to be equal to C squared. Now, 100 44 plus 25 comes out to 169. And if I go ahead and so 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 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 Soko Towa, 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 sign which is opposite over hypotenuse. So we'll have that the sign 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, 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 sign on the left side and the inverse sign on the right side that leaving me with just our angle X is equal to the inverse sign of 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 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 Soko Towa memory tool as well as this relationship with the Pythagorean theorem to solve 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 trigger 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 trick 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 eight and then the hypotenuse is the long side or 17. And 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 gonna get this cosine to cancel, giving us that B is the inverse cosine of 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. So we're going to say that 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 up 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 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. $x=30.000°,y=60.000°$

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 6500 ft to a scenic viewpoint in a canyon at an elevation of 4300 ft. We have that the path spans 4400 ft. And we're asked to determine the angle of inclination of the hiking path. OK. 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 was, is at an elevation of 6500 ft, which is going to be up here. So this distance would be 6500 ft. Now, we have that we're trying to reach a destination that is at an elevation of 4300 ft and that is gonna 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 4400 ft. So that means that this path to get to our destination is 4400 ft. 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 6500 ft. It's going to be some other height that we'll call h and the reason why is because 6500 ft 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 gonna be is gonna be the 6500 ft minus the 4300 ft because that's going to give us this distance, the 4300 ft of elevation to the mountain lodge. So 6500 minus 4300 will give us 2200 ft. 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 solver this angle using Sokoto Sokoto 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 we're, if we're looking at this angle, the opposite side is going to be 2200 ft. And then the hypotenuse is 4400 ft and the trigonometric function that uses the opposite and the hypotenuse is the sign. So we can see that the sign of our angle theta is equal to the opposite side of this triangle, which is 2200 ft divided by the hypotenuse, which is 4400 ft. Now to solve for our angle, theta, I'm going to take the inverse sign on both sides of the equation that's going to get the sign and the inverse sign to cancel on the left side leaving us with just the and theta is going to equal the inverse sign of 2200/4400. Now 2200/4400 is something that can be, this is equivalent to the inverse sign of one half because 2200 goes into 4402 times. So this would reduce to a half and the inverse sign 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. I hope you found this video helpful. Thanks for watching.