 Hello, and welcome to this screencast for Math 123. Today we're going to discuss how to use a reference arc for a given arc to find the sine and cosine values for that arc. So again, to do this we need to know the definition of a reference arc, and one thing we will assume in this screencast is that you are already familiar with this definition, and here is the definition for the reference arc, and what we're going to do now is to try to use this idea to find the sine and cosine values for 2 pi over 3. So for this we want to first find the terminal point of 2 pi over 3 so we can determine its reference arc. So remember our measurement starts at the point 1, 0, and as we've seen before, the arc 2 pi over 3 will have a terminal point about there, and the reference arc for that will be measured like that, and the value for that reference arc, x hat, will be pi minus 2 pi over 3, or pi over 3. So now if we draw in the point, or the terminal point of the arc pi over 3, we will have a point about there in the first quadrant, and in fact these two points will apply on the same horizontal line. In other words they have the same y-coordinate. Now one thing we know already are the x and y-coordinates for pi over 3. That comes from a previous section in the trigonometry book, namely section 1.3. This point, terminal point for pi over 3 has x-coordinate of one-half and y-coordinate of square root of 3 over 2. So what that means for 2 pi over 3 is that the y-coordinate will also be square root of 3 over 2, and due to the symmetry of the circle, and the fact that this is also pi over 3, the x-coordinates will actually be negatives of each other. So in other words for the point 2 pi over 3, the coordinates will be negative one-half square root of 3 over 2. And once we determine those coordinates of the terminal point, we know the cosine and sine values. The cosine is the x-coordinate, the sine is the y-coordinate. So we get cosine of 2 pi over 3 equals minus one-half and sine of 2 pi over 3 equals the square root of 3 over 2. If you want you can check that on your calculator. So in general, an arc and know what's reference arc, so in this case the reference arc is actually for each of the four points listed here is actually the value of t. The reference arc is kind of this arc right here and it's t. What these t's alongside here actually represent are the actual reference arc. Remember it's always measured positive. So each of these four points has a reference arc equal to t. And as you can see we're using the ideas of the four quadrants. The point here in the first quadrant has coordinates cosine of t, sine of t. And in the second quadrant because of that horizontal line, the y-coordinate is the same and the x-coordinate is negative. So we get negative cosine of t, sine of t. And as we proceed around the quadrants you can see the coordinates for all four points. The x-coordinates have the same absolute value. The y-coordinates have the same absolute value. They just take on the signs, the plus or minus signs depending upon which quadrant in which they lie. So in the third quadrant we get both negative coordinates and in the fourth quadrant the x-coordinate is positive and the y-coordinate is negative. Keep that in mind and it kind of says if you can find the reference arc and it comes out equal to t and you also know the cosine and sine values of t, then you can get the cosine and sine values for any of the four points shown on this diagram. So next problem. Determine the values of cosine of 11 pi over 6 and sine of 11 pi over 6. And I haven't saved a lot of room here and the idea is you can try this if you want first before proceeding. If you want to do that pause the recording now and come back after you have given tried to solve this problem. We will solve this on the next slide. Okay welcome back. The problem is stated there again and as with the other one our task is to find the terminal point of 11 pi over 6. That's the first task and so as we go all the way around the circle since this is in terms of 11 pi over 6 and we want to move general in a positive direction as we go around and we get down to this point here we've gone 3 pi over 2 but it might be nice to think of that as 9 pi over 6 so we can easily compare it to 11 pi over 6 and as then we go all the way around the circle and end up here it's at 12 pi over 6 and so 11 pi over 6 will have a terminal point oh let's say about there. So that's where we would find 11 pi over 6 and so now we need to determine the reference arc and remember the reference arc will be a positive arc and it goes to the nearest x intercept so there's the reference arc and as we try then to calculate the value of that reference arc we basically will get the final value minus the initial value is 12 pi over 6 minus 11 pi over 6 and so we get pi over 6 and so what we know now is that the cosine of 11 pi over 6 is going to be positive because the terminal point is here in the fourth quadrant so it will be equal to the cosine of pi over 6 in addition the sine of 11 pi over 6 because we're in the fourth quadrant again the y coordinate is negative this will be equal to minus the sine of pi over 6 and again we know the exact values for both cosine of pi over 6 and sine of pi over 6 and so now we can write the final values cosine of 11 pi over 6 will be the square root of 3 over 2 and the sine of 11 pi over 6 will be minus 1 half. So there you have two examples of how to use the reference arc to find the value of the sine and cosine values for cosine and sine for a given arc and I hope these two examples help you and that you will be able to do these problems on your own. Thanks and goodbye.