Hi, guys! I’m Nancy. And I’m going to show you how to solve a right triangle. Which I know pretty sounds pretty vague, like, what does that mean exactly? It just means find all the missing sides and angles. And it’s not that bad. Let me show you how to do it. Alright. Say you have a right triangle and you’re suppose to solve that right triangle. I know that’s a little unclear. Like I said, it’s a little vague as to what that means. It just means find any missing sides or angles that are not already labeled on your right triangle. And I’m going to show you how to do that. But first, just in case you were wondering if this is some kind of special right triangle.. like 30, 60, 90 or 45, 45, 90. I’m showing you this one on purpose because I know that 34 degrees is suspiciously similar to 30 degrees. But unless you have exactly 30 degrees as an angle or 60 degrees or 45 degrees, it’s not a special right triangle. Not so special. So we can’t use those. What can we do? Well, if you have a right triangle and you know at least one side and an angle that’s not the right angle, you can use a trig function to find the sides that you don’t know. And the first thing to do is to label those sides that you’re looking for that you don’t know. Give them names. They deserve that much, the dignity of having a name. And you can use whatever letters or variables you want. I’m going to use lowercase a and lowercase c. And I’m just using those because there’s this convention, this standard you might see where for the big A, the angle capital A, the side that’s directly across from it, a lot of times, it’s labeled little a, lowercase a. The side directly across from capital C is little c. And the side directly across from capital B angle is lowercase b, but we don’t need that because we already know what that is, 8. So I said that you could use a trig function to find each of these sides, but how do you know which trig function, sine, cosine, tangent to pick? Good question. Alright, so these are three of the main trig functions: sine, cosine, and tangent. This is a memory trick: SOH-CAH-TOA. It’s a mnemonic device meant to help you remember that sine of an angle is equal to the opposite side over the hypotenuse, cosine of an angle is adjacent over hypotenuse, tangent of an angle is opposite over adjacent. What are these opposite, adjacent, hypotenuse things I’m talking about? If you don’t know, you can go to my other trig video and jump to that. I do a whole introduction and explanation of that. But basically it depends what angle you’re focused on, what angle you’re looking from. We have this 34 degree angle in our triangle. From the point of view of that angle looking out, from it’s perspective looking out, this is the opposite side because it’s directly opposite that angle, directly across. Opposite side. The longest side is always the hypotenuse, the one that’s straight across from the right angle, the 90 degree angle. It’s always the hypotenuse. And then the other side that’s next to our angle but is not opposite and is not the longest side is the adjacent side. ADJ or A in our trick. So how do you know which of those three to pick? Well, say that we want to find this side, solve for a, here’s the rule of thumb. Pick the one of these three, pick the one that includes both the side that you want, that you’re looking for, that you don’t know, and the side that you know, something that you do know the number for, the value. So we want one that includes the opposite and the adjacent because we’re looking for a and we have 8. Opposite and adjacent, O and A, you may have already guessed, is tangent trig function; tan is opposite over adjacent, OA, the TOA in SOH-CAH-TOA. So all we have to do now is write a little equation. Tangent of 34 degrees, tan 34 degrees equals opposite over adjacent, which for us is a/8. Alright. So here’s your equation. You want to solve for a, so you want to get a alone. And right now you have this divided by 8 over there that you don’t want. So to undo that or get rid of that, you can multiply by 8, but both sides, multiply both sides by 8. OK. So the 8 in the bottom goes away over there and you just have a=8tan34, a=8tan34. And that’s really the answer, but you may to get an actual number like a decimal and round to the nearest tenth or something like that. So for that, you’ll need to plug this into your calculator, so either a scientific calculator or a graphing calculator. But make sure that you’re in degree mode when you use the tangent 34 degrees and not in radian mode, otherwise you’ll get weird number that’s not right. So in degree mode, when you plug in 8tan34, and if you round to the nearest tenth, it’s…a is approximately, roughly 5.4. So a is about 5.4, that opposite side is about 5.4 long. OK. So you have two sides of the triangle and you don’t have the third, there’s one side left that you haven’t solved for. If you have two sides and you want the third, you have a choice in how you find it. You can either use a trig function or you can use Pythagorean theorem. So you can pick your poison, whichever one you want. I’m going to show you both. Using the trig function, it’s the same idea as before. If you want to find this side c, pick the trig function that includes both that side, which is the hypotenuse, and the side that you do know the number for the value of, which is adjacent. So you want the one that includes hypotenuse and adjacent, which is cosine because cosine is adjacent over hypotenuse. So now you just need to write an equation like you did before that’s cosine 34 degrees equals adjacent over hypotenuse, which is 8/c. OK, and you want to solve for c. It’s down there in the denominator in the bottom, which is different from the one before. How do you solve when the variable you want is down in the denominator? There’s an extra step. You want to get rid of it in the denominator and bring it up top. And the trick is multiply both sides by c so that you clear it over there and bring it up on one side. So multiply both sides by c. So we’re closer. You have c here. But you also have this times cosine 34 degrees. You want to get c alone so anything that’s attached that you don’t want there, divide out. So divide both sides by cosine 34 degrees. So c=8/cos34. Again, just put this in your calculator using degree mode, and if you round it, it’s roughly 9.6. OK. So there’s another way to find that last side. Like I said, you could use the Pythagorean theorem. OK. So here’s the Pythagorean theorem or pythag as the cool kids say. Just don’t call it the “pie-tha-gorean thee-orem” like I did when I was little, it’s definitely wrong. Definitely weird and the wrong emPHAsis for the sylLABles. But basically it’s a²+b²=c², and hopefully you’ve seen it before. c is always the hypotenuse, and a and b are the legs of the triangle that are on either side of the right angle. If you have two of them, you can find the other. And we have b. We did find a earlier with our tangent. It was about 5.4. And we want c. So we can use the numbers that we found… we have our a and b, and plug them in, and solve for c. And it’s better to use the unrounded decimal, by the way, which hopefully you still have in your calculator because that will give you a more accurate answer in the end. So when you plug it in, it looks like this, and this, if you square it, it turns out to be 29 point something plus 8² which is 64. So all together, it will look like this.. So it’s 93 point something, something, something equals c². And to get just c, you can square root both sides to get c alone. And use the positive answer since this is a length, a distance that should be positive. And when you do that, c is about 9.6, which we knew already. So that’s how you can use the Pythagorean theorem. And found that the hypotenuse is about 9.6 long. OK, so you found all the sides of the triangle. I know you may think you’re done but there’s one last thing to figure out and clearly label, and it’s this other angle because it’s not already clearly labeled. Don’t worry, this is the fastest, easiest part because in a triangle, all the angles inside have to add up to 180 degrees. So if we already know for sure that this is 34 degrees, and this angle here is 90 degrees because it’s a right angle, we know that the 90 and 34 and whatever this other angle is, better equal 180 degrees. And we can write an equation that says that. So here’s the equation, all three angles, including angle B, which we’re looking for, have to equal 180. If we clean this up, those two together, 124. And if you subtract that from both sides, you’ll see that angle B has to be 56 degrees. So that angle is 56 degrees and the triangle is totally solved. All sides and angles are labeled, and you’re done. Just a few things I want to mention. If you got a different side given to you in the beginning, like if you weren’t told this side, but you were instead given the hypotenuse and no other sides, don’t worry, it’s the same kind of steps. Just make sure you pick a trig function that includes both that hypotenuse you know and whatever side you’re looking for. So it’s the same idea. Also, if you are given no angles in the beginning other than the 90 degree angle, you may need to use inverse trig functions to find an angle and fully solve. And that’s a whole other video if you see that. And one last thing, don’t worry if you see a triangle given to you that’s, like, rotated, it’s oriented differently. No matter what orientation you get, it’s the same idea. Just focus on an angle, figure out for yourself which side is opposite, which side is hypotenuse, which side is adjacent to it, and then pick a trig function knowing what you have and what you need. So it’s the same kind of steps. So I hope that helped you understand how to solve right triangles with trig. I know that trigonometry is everyone’s favorite… It’s OK.You don’t have to like math.. But you can like my video! So if you did, please click ‘Like’ or subscribe.