 So early on in our lecture series we learned about two different ways of defining the six trigonometric ratios. One way is to define them using a point of some kind. So we have the x-axis, the y-axis, and in which case then we take some point which terminates in the plane. If we take the ray that emanates from the origin passing through this terminal point, then this forms an angle, of course. And we can define the six trigonometric ratios with respect to this point. That is, you're going to use the x and y coordinates of that point. Let r be the distance this point is from the origin. And so you can define the six trigonometric ratios using the numbers x, y, and r. Well, of course, if this point is in the first quadrant, then naturally a right triangle can be connected to this angle. And so the sine, cosine, tangent, etc. of this angle, whether you take this right triangle perspective, it'll be the same trigonometry that you get. As opposed to this terminal point perspective. But the right triangle approach has what seems like an obstacle here, right? The angle sum of a triangle can never exceed 180 degrees. And if there's a right angle, that means 90 degrees is what we're accounted for, this angle theta right here must be acute. You can't get something obtuse. You can't get something that terminates in the second quadrant or anywhere else, mind you, right? But this idea of an initial point also works there as well. So it feels like the right triangle approach to trigonometry is limited. But it turns out that's actually not the case with the use of reference angles. We actually could extend right triangle trigonometry to any quadrant we want. What if, for example, our angle actually terminates in the second quadrant? And so the angle is itself an obtuse angle, right? There's no triangle that could have a right angle and obtuse angle. So it feels like right triangle trigonometry is not going to work here. Nope, it actually does. What if we consider the reference angle, which theta hat, which we talked about previously, using theta hat, we can produce a right triangle whose angle in question is the reference angle of theta and up to a sine difference that is up to a plus or minus. The sine, cosine, all the trigonometric ratios will be the same in this situation. Because after all, if you have this point, x, y, and then the distance here is r, well, if the x-coordinate gives us this point, then the length of this side, the adjacent side is going to be absolute value of y. The other side, the opposite side, its length is going to be absolute value of y. And then the distance r is always positive in this situation here. So if you do the trigonometry of this triangle, you're going to get that sine of theta hat is equal to the absolute value of y over r, which this is equal to the absolute value of sine of theta. And that's because sine is equal to the absolute value of y over r. Like, you know, sine is just y over r, so we're just taking the absolute value there. Similarly, if you were to do cosine of theta, right, you're going to end up with the absolute value of x over r there, which is the same thing as the absolute value of cosine of theta, given that cosine is x over r. If we take its absolute value, those things are going to be the same thing. So ignoring the algebraic sine plus or minus, sine cosine of the reference angle will be the same as sine and cosine of that given angle. That's also true for tangent, cotangent, secant, and cosecant as well. And this is not just true for the second quadrant. This will be true for the third quadrant, the fourth quadrant. If your angle is larger than 360, less than zero degrees, it doesn't matter. We can use reference angles to compute trigonometric ratios of that right triangle. So therefore, right triangle trigonometry extends to any angle measure whatsoever as long as we use reference angles. And so what we've now just basically proven is the so-called reference angle theorem, which tells us that a trigonometric function of an angle has the same absolute value as the trigonometric function of the reference angle. Because that is, they might differ by plus or minus one as a sine, right? And so we always get, we always get this idea right here. Sine of the reference angle will equal the absolute value of sine of theta. They could equal if they're both positive, because the sine and cosine of the reference angle will always be positive, but they could also be off by negative sine depends on the quadrant. So as long as we can keep track of the quadrant, then we can use reference angles to compute various trigonometric ratios. So let's keep track of things for us here. Like I mentioned before with this picture, R is always going to be positive because it's the distance between the origin and a point. That's always a positive number. The X coordinate could be positive or negative. The Y coordinate could be positive or negative. And so in general, if you're in the first quadrant, X is positive, Y is positive. In the second quadrant, you get that X is negative, Y is positive. In the third quadrant, you get that X and Y are both negative. And in the fourth quadrant, you're going to get that X is positive and Y is negative. Now, since sine is just going to be the Y over R, R is always positive, the sine of sine, right? Be careful of that. S-I-G-N of S-I-N-E will always be the same as the Y coordinate. And likewise, the sine of cosine will always be the same as the X coordinate. So if we keep track of the quadrant, then we can keep track of the sine of our trigonometric function. So for example, in the first quadrant and second quadrant, sine will be positive because it's above the X-axis. In the third and fourth quadrant, sine will always be negative because it's below the X-axis. A cosine on the hand is positive in the first quadrant and in the fourth quadrant because that's the right of the Y-axis. And cosine is negative in the second and third quadrant because it's to the left of the Y-axis. The other four trigonometric ratios, we can keep track of sines based upon their ratios. Tangent is sine over cosine. So if you take a positive divided by a positive, you get a positive. If you take a positive divided by a negative, you get a negative. If you take a negative over a negative, it's a double negative. You'll get positive. If you take a negative divided by a positive, you're going to get negative. And be aware that the reciprocal of tangent is cotangent, so it's going to have the exact same sines. So tangent and cotangent will be positive and negative at the same spots. Likewise, as cosecant is the reciprocal of sine, it'll have the same sines there. And secant is the reciprocal of cosine, so they'll have the same thing as there. Because you're just flipping the fraction upside down and it doesn't change its sine whatsoever. So let me show you how you can compute trigonometric ratios using reference angles. So let's find the exact value of sine of 240 degrees. You'll notice that 240 degrees, if we think about that one, it's larger than 180 degrees, but smaller than 270 degrees. So this lives in the third quadrant. So to find the reference angle, so this is our angle theta right here, to find the reference angle in the third quadrant, we're going to take our angle theta and subtract from it 180 degrees. So we take 240 degrees minus 180 degrees. This is going to give us 60 degrees. Now, this is one of the angles we're supposed to remember. So sine of 240 degrees, this is going to equal sine of its reference angle, 60 degrees, plus or minus, right? It's one or the other. Now, if you're in the third quadrant, sine is negative. So sine of 240 degrees is equal to negative sine of 60 degrees. And sine of 60 degrees, as we've learned earlier, this is one we should have memorized, sine of 60 degrees is the square root of 3 over 2. So we see that sine of 240 degrees is negative root 3 over 2. And this is a calculation we're able to do without a calculator because it involves an angle which references one of our special angles. Let's do another example. Let's find the exact value of tangent of 315 degrees. Well, if theta is equal to 350 degrees, notice it's not quite 360, but it is larger than 270 degrees. So this is something that lives in the fourth quadrant. If you're in the fourth quadrant, the reference angle is how much closer can you get to be 360 degrees? So theta hat, the reference angle will be 360 degrees minus theta. So if you take 360 and you subtract from it 315, you end up with 45 degrees as our reference angle, like so. And so the other thing also pay attention to here is if we're looking for tangent here of 315 degrees, this is going to equal tangent of 45 degrees, again, plus or minus. Which quadrant are we in? We're in the fourth quadrant. So in the fourth quadrant, you get that cosine is positive, sine is negative. So the quotient tangent, which is sine over cosine, the net will be a negative actually. And so we get that tangent 315 degrees is negative tangent of 45 degrees. Well, how do you do that? Well, tangent to 45 degrees is sine of 45 degrees over cosine of 45 degrees. So we get negative sine of 45 over cosine of 45. Well, sine of 45 degrees is root 2 over 2. Cosine of 45 degrees is also root 2 over 2. And so you get the same quantity divided by itself. This simplifies just to be negative 1. So we see that tangent of 315 degrees is going to be negative 1. Consider cosecant of 300 degrees. The first thing to do is to compute the reference angle. If theta is equal to 300 degrees, again, 300 is larger than 270, but smaller than 360. This will again be in the fourth quadrant. So the reference angle is computed by taking 360 minus 300. So our reference angle is going to be 60 degrees right here. All right. The next thing to consider is then the sine. So cosecant of 300 degrees. This is going to equal cosecant of 60 degrees plus or minus. Which one is it? Positive or negative? Well, which quadrant are we in? We're in the fourth quadrant. So again, in the fourth quadrant, cosine is positive. Sine is negative. So since sine is negative cosecant, which is just one over sine, will likewise be negative in this quadrant. So let's then mark that up. So we get that cosecant of 300 degrees is negative cosecant of 60 degrees. Well, cosecant is just going to be 1 over sine of 60 degrees. And sine of 60 degrees is, of course, square root of 3 over 2. So if we take the reciprocal of this, we end up with negative 2 over the square root of 3, which then is cosecant of 300 degrees. Now, if you are insistent on rationalizing the numerator, excuse me, the denominator, you can do that. It's really not necessary. But if you times the top and bottom by the square root of 3, like so, you could also write the answer as negative 2 square root of 3 over 3. That is also correct. But I think negative 2 over square root of 3 is also, definitely also correct. The two numbers are equivalent to each other. And there's probably no need to rationalize the denominator. But if you feel compelled to do so, satisfy your compulsion and do it right now. Let's do one last example before we end lecture 6 right here. Let's compute cosine of 495 degrees. Well, notice that 495 degrees is larger than 360 degrees. So let's compute something coterminal to it. If I subtract 360 from 495 degrees, we're going to actually end up with 135 degrees. So cosine of 495 degrees is just cosine of 135 degrees. So you can always replace an angle with something coterminal to that. And you'll have the exact same trigonometric ratio there. So let's think about the reference angle now, because still 135 degrees is not the first quadrant. If theta equals 495, well, again, that's equivalent to 135. So we're good with that. That's in the second quadrant, like I mentioned. It's bigger than 90 degrees, but less than 180 degrees. So the reference angle, if you're in the second quadrant, think about that. Your angle's terminating over here. Your reference angle will be that angle right there. So how much more do we need to get to 180? So you're going to take 180 minus theta. And so we take 180 degrees minus 135 degrees. And that gives us 45 degrees, our reference angle. So continuing on here, we get that cosine of 495 degrees. This is equal to cosine of 135 degrees, which is equal to plus or minus cosine of 45 degrees. Again, pay attention to the quadrant. If you're in the second quadrant, we get that cosine is negative, sine is positive. So we actually need to get that. We're going to get that cosine is negative on this one, like I said. So we're going to get negative cosine of 45 degrees. Cosine of 45 degrees is root 2 over 2. And therefore cosine of 495 degrees is negative root 2 over 2. And we were able to do this without the use of any calculator whatsoever because we use reference angles, quadrants. And in these cases, everything turned out to be one of the special angles, 0, 30, 45, 60, or 90 degrees. And so in those settings, which are very common in a trigonometric course, we can do these calculations without a calculator whatsoever. That brings us to the end of lecture six. Thanks for watching, everyone. I hope you learned something. If you did, please give this video or any of the videos you're watching a like, click the like button there to see more videos like this in the future. 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