Special Right Triangles - Video Tutorials & Practice Problems

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45-45-90 Triangles

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Welcome back everyone. So up to this point, we've spent a lot of time talking about trigonometric functions, the Pythagorean theorem and how they all relate to the right triangle. Now, what we're going to be learning about in this video is some of the special and common right triangles that you're going to see. And in this video specifically, we're going to be talking about the 45 45 90 special tri. Now, the reason that these triangles are special is because it turns out that because they show up relatively frequently, there are actually some shortcuts that you can use to solve these triangles very fast. So if you don't like all the brute force work we've been doing with trigonometric functions of the Pythagorean theorem, you're going to learn some shortcuts for solving these triangles in this video. So without further ado let's get right into things. Now, when you have a triangle with 45 degree angles like this triangle down here, for example, this is going to be a situation where you have the special 45 45 90 triangle. And in these triangles, the two legs of the triangle are always going to be the same length. So if you ever see a situation where you have a right triangle and two of the legs are the same, that means you're dealing with this special triangle. Now, what we can do with this is we can actually solve for the hypotenuse of the triangle by simply taking a multiple of the lake length. And the multiple that you're going to look for is the square root of two. Because if you take a leg like five and you multiply it by the square root of two, this will give you the hypotenuse. And that's the answer we just solved for the long side of this triangle. So as you can see this shortcut right here, solving for the sides of the triangle really straightforward and really fast. Now, if you didn't remember this relationship, there is another strategy you can use, which is simply the long version of using the Pythagorean theorem. So let's say that we set this side to A, that side to B and then the hypotenuse equal to C. And we want to solve for the hypotenuse. Well, you could say that A squared plus B squared equals C squared. That's the paging theorem. And in this case, we said A and B are both. So we have five squared plus five squared is equal to C squared and five squared is 25. So we have 25 plus 25 equals C squared, 25 plus 25 is 50. And what we can do is take the square root on both sides of this equation to get that C is equal to the square root of 50. And the square root of 50 actually simplifies down to five times the square root of two. So notice when using the long version of this problem solving, we get to the same answer. But this is what's nice about the shortcut is it lets you get this answer without having to go through this long process. Now, to ensure we know how to solve these types of triangles, let's see if we can solve some examples where we have this special case. So for each of these examples, we're asked to solve for the unknown sides of each triangle. And we'll start with example a now notice we have 245 degree angles and two legs that are the same length. So that means we're dealing with a 45 45 90 triangle. Now recall that if we want to find the missing side or the hypotenuse, we just need to take one of the legs and multiply it by the square root of two. Well, one of the legs are 11 and then we multiply this by the square root of two. And that right there is the answer. Notice how quick it is using this method. See it's very straightforward and that's what's really nice about these special cases. But now let's take a look at example, B in this example, we have a 45 degree angle and we are given the hypotenuse. So how could we go about solving this? Well, first off, we need to figure out if we actually are dealing with a special case triangle. And it turns out that we are because since we have a 45 degree angle here and a 90 degree cusp there, we know by default, this has to be a 45 degree angle. You see all the angles in a right triangle have to add to 100 8090 plus 45 plus 45 equals 180. So this is a special case triangle. Now to solve for the missing sides, what we can do is use this relationship. Notice in this situation, we're given the hypotenuse or the long side. So what I'm going to do is take the hypotenuse instead of equal to the number we have, which is 13. And I'll say that that's equal to the leg multiplied by the square root of two. Now, to solve for the leg, what I can do is divide the square root of two on both sides of this equation. That'll get the square root of twos to cancel, giving us that the leg of this triangle is equal to 13 over the square root of two. Now, what I can do is rationalize the denominator here by multiplying the top and bottom by the square root of two, that'll get these square roots to cancel, giving us that the leg of this triangle is equal to 13 times the square root of 2/2. So what we're gonna end up with is 13 radical 2/2 for this side of the triangle and then 13 radical 2/2 for that side of the triangle. Because again, for a 45 45 90 triangle, these two sides have to have the same length. So that is how you can solve 45 45 90 triangles. And this is the shortcut that you can use. So hope you found this video helpful. Thanks for watching and let me know if you have any questions.

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Problem

Problem

Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

A

$x=81$

B

$x=9\sqrt2$

C

$x=2\sqrt9$

D

$x=\sqrt{162}$

3

Problem

Problem

Without using a calculator, determine all values of P in the interval $\left[0\degree,90\degree\right)$ with the following trigonometric function value.

$\csc P=\sqrt2$

A

$P=30\degree$ only

B

$P=45°$ only

C

$P=60°$ only

D

$P=30°,60°$

4

concept

Common Trig Functions For 45-45-90 Triangles

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Hey, everyone. So in a recent video, we talked about the special 45 45 90 right triangles and we learned how you can use some shortcuts to quickly find the missing sides. Well, in this video, we're going to be learning about the common trig functions and how they are associated with the 45 45 90 triangle. And we're going to see if we can recognize any kinds of patterns for these trig functions. Because what we're going to learn for these special cases is there may be shortcuts when finding ratios for the trigonometric functions as well, meaning these triangles could be overall a lot easier to solve. So I'm always interested in finding shortcuts and hopefully, you are too. So let's get right into this. Now, we're going to start with the sine function and we know that sign is the option divided by the hypotenuse. Now, if we go to a 45 45 90 right triangle, we can go to either one of the two angles because they're both 45 degrees. So if we take a look at this angle, for example, and we go to the opposite side, we can see that the opposite is five. And if we divide this by the hypotenuse or along side, that's going to be five times the square root of two, we can cancel the fives here giving us one over the square root of two. And by rationalizing this denominator, I'll multiply the top and bottom by the square root of two. This will get the square roots to cancel giving us square root of 2/2. So that means for the sign, we'll end up with square root 2/2. But now let's try solving for the cosine for the cosine, we get adjacent, divided by hypotenuse. Now, if we go to one of our 45 degree angles, the adjacent side is five. And if we divide this by the hypotenuse, we'll have five times the square root of two, the five will cancel, giving us one over the square root of two. And we already figured out this is the same thing as square root 2/2. So this will just simplify to radical 2/2, which is the cosine for this right triangle. Now, let's take a look at the tangent for the tangent we have opposite over adjacent. So what I'm going to do is go over here to our right triangle, look at one of our angles. And if I go to the opposite side, we end up with five, if I go to the adjacent side, we also have five So we'll have 5/5, which is simply one, meaning the tangent for our 45 45 90 triangle is one. So let's see if we notice any patterns with these trigonometric functions. Well, something that I noticed is that the sine and cosine are the same and the tangent just comes out to one. So this is something that's pretty straightforward about these triangles is that you get the same sine and cosine values. But let's take a look at these other reciprocal identities like the co sequent sequent and cotangent. So for the code secret, we know that this is just one over the s of data, which means all we're going to do is flip this fraction that we got. So we'll have two over the square root of two. And then we can rationalize this denominator when doing this, the square roots will cancel on bottom of the fraction giving us two times the square root of 2/2. These are going to cancel right here, meaning all we're going to end up with is the square root of two for the cosecant. And because the sine and cosine were the same, that means one over the cosine, which is our second is also going to be the square root of two. So all we now have to solve for is the cotangent, which is just one over the tangent. But we learned that the tangent is one. So the cotangent will be 1/1, which is simply equal to one. So notice how really straight for all these trigonometric functions are, when you're looking at a special case 45 45 90 triangle, we can recognize that the sine and cosine will always be square root 2/2, that the cosecant and see it will always just be the square root of two and that the tangent and cotangent will always be one. So these are all of the common trig functions for the 45 45 90 triangle. Hope you found this video helpful and thanks for watching.

5

concept

30-60-90 Triangles

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5m

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Welcome back everyone. So in recent videos, we've been talking about special case right triangles and we recently learned about the 45 45 90 triangle and some shortcuts you can use for solving this special case. Well, in this video, we're going to be taking a look at the 30 60 90 triangle, which is another special case. And for these triangles, there are shortcuts you can use to solve them as well. And this is a type of triangle that is going to show up a lot, not only in this course but also in future math courses and possibly other courses too like physics or other science related courses because this situation just tends to happen a lot. So without further ado let's get right into some shortcuts, you can use to solve this triangle. Now, the thing that's special about this triangle is that you can relate the side lengths to the shortest leg on the triangle. Now keep in mind for the 30 60 90 triangle, it's a bit different than the 45 45 90 triangle because for the 45 45 90 case, we learned that two of the side lengths are the same for the triangle. And so you can use one relationship to relate all the sides. Well, for the 30 60 90 case, the two side lengths are not the same, we have two different lengths. So what we need to do here is keep track of which side relates to which other side. But we can do that using the shortcuts. So for example, if you want to find the hypotenuse of a 30 60 90 triangle, you can take the short leg of your triangle, the shortest side and you can multiply it by two. And if you want to find the long leg of triangle, you can take the short leg of the triangle again, and you can multiply it by the square root of three. Now again, I want to emphasize these shortcuts only work if you're dealing with a 30 60 90 triangle. And this is the special case we have in this example. So let's see if we can solve for the missing sides. Well, I can see that the hypotenuse is going to be the shortest leg multiplied by two. I can see the shortest leg is five. And if we multiply that by two, we're going to get 10, meaning the hypotenuse is 10. Now, I can see that the long leg of the triangle is equal to the short leg which we have down here. And we said that that's five multiplied by the square root of three So the long leg of the triangle is going to be five times the square root of three. So this is how you can find the missing sides of a 30 60 90 right triangle. So as you can see, it's still very straightforward when you have these shortcuts available, but to make sure we know how to use these shortcuts. Well, let's try some other examples that are a little bit more complicated. So for these examples, we're asked to solve for the unknown side of each triangle. And we're going to start with example a. Now, in this example, I see that we have the hypotenuse and a 30 degree angle. So what we first have to ask ourselves is, is this a 30 60 90 triangle? Well, it actually is because if this is 30 degrees and this is 90 degrees, then the other missing angle has to be 60 degrees because 60 plus 30 gives you 90 then you add another 90 you get to 180 all angles in the triangle have to add to 180 degrees. So this is a special case. So that means we can use the shortcuts we learned about, well, first I'm going to try solving for the short leg and I'm going to do this using this relationship up here because we have that the hypotenuse is equal to the short leg multiplied by two. Now, the hypotenuse we can see is the longest side which is eight. And so that's going to equal the short leg of the triangle multiplied by two. So if I want to find the short leg, all I need to do is divide both sides of this equation by two, that's going to get the twos to cancel on the right side, giving us that the short leg of our triangle is equal to eight divided by two, which is four. So that means this missing side is four. Now, if I want to find the long side of the triangle, I could use the Pythagorean theorem from here or I could use this relationship that says that the long leg is the short leg multiplied by the square root of three. So we have that the long leg is equal to the short, I'm just gonna write it as long, by the way, we have that's equal to the short leg, which we just figured out is four and that's multiplied by the square root of three. Meaning this missing, the triangle is four times the square root of three. So that is how you can use shortcuts to find the missing sides of a 30 60 90 triangle. But now let's try another example in this case example. B and for this example, we're asked to find the missing sides as well. And what I can see is that we have a 60 degree angle and the long side of the triangle given to us now because we have a 60 angle here. By default, this other angle has to be 30 degrees because all angles of the triangle have to add to 180. So this is a special case. So let's try using some of these relationships. Well, since I see that we have the long leg, I'm going to use this relationship right here, which is that the long leg is going to be equal to the short leg multiplied by the square root of three. Now, I can see that we have that the long leg of the triangle is one, so we'll have one, the long side is equal to the short side multiplied by the square root of three. So what I can do from here is divide the square root of three on both sides of this equation, which will give the square root of threes to cancel on the right side, giving us that the short leg of this triangle is equal to one divided by the square root of three. And I can go ahead and rationalize this denominator by multiplying the top and bottom by the square root of three. They'll get the square roots to cancel, giving us that the short leg of this triangle is equal to the square root of 3/3. So that's going to be the short leg of the right triangle. Now, our last step for solving this right triangle is to find the hypotenuse or the longest side. But I can see based on the relationship up here that the hypotenuse is going to be the short leg, which we already determined is square root 3/3, multiplied by two. So that means that this missing side of the triangle is going to be two times the square root of 3/3. So this is how you can find all the missing sides of a 30 60 90 triangle using some simple shortcuts. So that's how you solve these types of problems. Hope you found this video helpful. Thanks for watching.

6

Problem

Problem

Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

A

$x=10,y=5\sqrt5$

B

$x=3,y=4$

C

$x=5\sqrt3,y=10$

D

$x=5,y=5\sqrt2$

7

Problem

Problem

Without using a calculator, determine all values of A in the interval $\left[0,\frac{\pi}{2}\right)$ with the following trigonometric function value.

$\cos A=\frac{\sqrt3}{2}$

A

$0$ only

B

$\frac{\pi}{4}$ only

C

$\frac{\pi}{6}$ only

D

$\frac{\pi}{3}$ only

8

concept

Common Trig Functions For 30-60-90 Triangles

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6m

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Welcome back everyone. So up to this point, we've been talking about special case right triangles and we recently took a look at the 30 60 90 case. Well, in this video, we're going to be taking a look at the trigonometric functions associated with the 30 60 90 triangle. Now, so far, the trigonometric functions we've learned about have been a bit tedious just because we have to find ratios for each of the situations and see how they relate to each other. But what we're hopefully going to figure out in this video is that for the special case triangles, there are patterns that show up when you solve for the trigonometric ratios and functions. So without further ado let's get right into this and see if we can find some shortcuts and patterns that show up. So we're going to start with finding the sign of a 30 60 90 triangle. And we're going to focus on the 30 degree angle for these trigonometric functions on the left side. Now recall that sine is opposite over hypotenuse. So if I go to a 30 degree angle, the opposite side of this triangle is five and the hypotenuse or the long side of the triangle is 10 and 5/10 reduces to one half, meaning that the sign of 30 degrees is one half. But now let's take a look at the cosine of our 30 degree angle for the cosine, it's going to be adjacent over hypotenuse and the adjacent side to the 30 degree angle is five times the square to three. And then this is divided by the hypotenuse which is 10 and again, 5/10 reduces to one half. So we'll end up with one times the square root of 3/2 and one times square to three is just square to three. So all we're gonna end up with is the square root of 3/2. And that's our cosine. But now let's take a look at the tangent for the tangent. It's going to be opposite over adjacent. Well, going to this triangle, we can see that the opposite side is five and then the adjacent side is five times the square root of three. Again, the fives will cancel, giving us one over the square root of three. And if I go ahead and rationalize this denominator by multiplying the top and bottom by the square root of three, we'll get the square roots to cancel, giving us the square root of 3/3 as our tangent. So this is what we get for the sine cosine and tangent of our 30 60 90 triangle. When specifically looking at the 30 degree angle. But now let's solve for the cosecant. And we're going to use reciprocal identities to solve these next three trigonometric functions. Because I noticed for the cosecant, it's the same thing as one over the sin of theta. And we said that the sin of theta is one half. So all we need to do is flip one half which is just going to be 2/1 or two. So the cosecant of our 30 degree angle is two. Now let's take a look at the second of our angle. Well, for the second, we can see that this is one over cosine. And we always said that the cosine is square root 3/2. So for the second, we're going to get two over the square root of three, which I will need to rationalize this denominator. So multiply the top and bottom by the square of three, cancel the square roots giving us two times the square root of 3/3, which is the result for our second of 30 degrees. But now let's take a look at our cotangent. The cotangent is going to be one over the tangent. And we figured out that the tangent is square root 3/3. So the code tangent is going to be three over the square root of three just flipping this fraction. And again, I'll need to rationalize this denominator. When doing this, the square roots will cancel giving us three times the square root of 3/3. And these three are going to cancel leaving us with just the square root of three. So that is how you can find the trigonometric functions for the 30 degree angle. Now, because this is a 36 90 triangle, we also need to find all the trigonometric functions for the 60 degree angle. So let's go ahead and do that on this right side. So for the sine function, if we're looking for 60 degrees, recall that this is opposite over hypotenuse. Now, the opposite side is five times the square root of three. And this is going to be divided by the hypotenuse which is 10, the five and 10 will reduce to give us just a two in the denominator. Meaning that all we're going to have is the square root of 3/2 for our sign. Now let's take a look at our cosine for the cosine of 60 degrees, we're called that this is adjacent over hypotenuse. So we're going to go to the adjacent side of the 60 degree angle which is five and then divide this by the hypotenuse which is 10, 5/10 reduces to one half, meaning that the cosine over a 60 degree angle is one half. Now let's take a look at the tangent which is opposite over adjacent the side opposite to 60 degrees is five times the square root of three and then adjacent side is five, the files will cancel, giving us just the square root of three, meaning that the square root of three is the tangent of 60 degrees. Now, for these last three trigonometric functions, we're going to use the reciprocal identities. Now, Kent is the same thing as one over S sign. So if I want to find the coin of 60 degrees, I just need to flip the sign of 60 degrees which would be two over the square root of three. Now, I can go ahead and rationalize this denominator, that'll get the square roots to cancel on the bottom, giving us two square root 3/3. So that is going to be the cosecant of our 60 degree angle. Now, what I can also do is find the second of our 60 degree angle and to do this, what I need to do is it's one over cosine. So I need to go ahead and flip the cosine of 60 degrees since the cosine of 60 degrees is one half, that means that the sequence of 60 degrees is going to be 2/1, which is just equal to two. So two is the sequin of 60 degrees. Now, lastly, we're going to take a look at the cotangent of 60 degrees and all I need to do is go ahead and flip the tangent that we got here. So we said the tangent is squared to three. So the cotangent is going to be one over the square root of three. I can go ahead and rationalize the denominator getting the square roots to cancel on bottom, giving the square root of 3/3 for the cotangent of 60 degrees. So these are all of the trigonometric functions evaluated for the 30 60 90 triangle. And you may notice that we have some patterns that emerge when we figure out what all these trigonometric values are because notice how the sine and cosine will switch places when looking at the different angles because the sine of 30 is one half, whereas the cosine of 30 is square root 3/2. But when we go to the 60 degree angle, the sine is square root 3/2 and the cosine is one half, we see the same kind of behavior in the coin and sequin since these are reciprocal identities, you just switch them when looking at the different angles. Now we also see this behavior in the tangent and code tangent because notice when looking at the 30 degree angle, the tangent is square root 3/3 and the code tangent is square root three. But when looking at the 60 degree angle, the tangent is just square root three and the code tangent is square root 3/3. So we see the same kind of switch when looking at this trigonometric function. So hopefully, this helps you to recognize some of the patterns and give you some general understanding as to how you can find the trigonometric functions for a 30 60 90 triangle. I hope you found this video helpful. Thanks for watching.