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Hi guys! Im Nancy. And Im going to show you how to remember the unit circle in trigonometry. What is the unit circle? Well it just means a circle with a radius of one. Thats all. Sounds simple… but actually there are a lot of angle values and sin and cos coordinates that you end up having to memorize. So knowing the unit circle is kinda a necessary evil. But dont worry! Im going to show you some patterns and tricks that will help you remember it. OK. So lets take a look at this. This is our unit circle. These are our axes. This is the x-axis. And this is our y-axis. These two lines touch the circle at 4 points. The right. The top. The left. and the bottom. When youre finding an angle on the unit circle we always start here on the right side and we move up counter-clockwise we rotate to get to angles. All the way around. So lets look at the 4 angles you get at this point this point this point and this point. So this first angle is 0. Because thats where were starting. Then if we move up and rotate counter-clockwise through a quarter of the circle we have the angle Pi/2. These angles are in radians, not degrees. They have Pi in them. Then if we move another quarter of the way through the circle we have the angle Pi. This is half the circle. So the whole circle will be 2 times that. 2Pi. Then if we move another quarter of the way through the circle we have the angle 3Pi/2. One quarter through the circle was Pi/2. So three quarters of the way around the circle is 3Pi/2. OK, finally, if
we move another quarter circle we get 2Pi. Now weve come full circle. And notice this angle here, this position is 0 but also 2Pi. OK. Now lets look at the Pi/4 angles. Those other angles we just did divided the circle into 4 equals parts. These are going to divide it again. So if one quarter was Pi/2, if we divide it again by 2 we get Pi/4. And these ones are always in the exact center of the quarter circle. The first one is 1Pi/4. Just Pi/4. Next one is 3Pi/4. Then 5Pi/4. Then 7Pi/4. Notice that pattern: 1, 3, 5, 7. So as you can see this is rapidly turning into a disaster. Now we have the Pi/6s. So if one quarter of the circle is Pi/2 a third of that, if you divide by 3, is Pi/6. So the first “over 6” is Pi/6. Then if we mirror it on the other side of the axis we have 5Pi/6. Down here 7Pi/6. And down here 11Pi/6. OK. Finally we have the Pi/3s. The first Pi/3 is here. It might seem wrong to you that Pi/3 is bigger than Pi/6 but remember youre dividing by those numbers. Youre dividing by 3 here, but youre dividing by a bigger number, 6, here. So this is actually a smaller, lower value than that. Anyway, this is Pi/3. If you reflect it across the y-axis, you have 2Pi/3. Down here is 4Pi/3. And down here is 5Pi/3. Alright. Now let me show you what the unit circle is actually good for. And that is knowing the cos, sin and tan values of each of these angles. Each of these angles that we drew in touches the circle at a point. So let me show
you some of these points. Alright. So every angle on the circle touches the circle at a point. An (x, y) point. It has an x-coordinate and a y-coordinate. So for this angle: 0. The x-coordinate is 1, because this is a unit circle. It has a radius of 1, so the x distance is 1. And the y-coordinate is 0, because this point has no height. The distance in the y direction is 0. Now this is very important these two coordinate values also give you your cos and sin. The first value, the x-coordinate, is always your cos. And the second value, the y-coordinate, is always your sin. So, for this angle of 0, the cos is 1 and the sin is 0. Now lets move to the top of the circle at Pi/2. The x distance is 0. And the y distance is a full 1 unit. So we have (0, 1). The cos is 0. The sin is 1. On the left side we have (-1, 0) as (x, y) coordinates. On the bottom of the circle we have (0, -1). Alright, so lets fill in the points for the other angles in the circle. All you need to remember are these 3 numbers. You probably wish they were nice, neat whole numbers but theyre not. They have square roots and fractions in them. Just remember: 1/2 is the smallest number sqrt(2)/2 or “root-2-over-2” is the middle size number and sqrt(3)/2 is the largest number. Youre gonna be using these again and again and again. So lets look at the angle Pi/6. For the angle Pi/6 that point has an x distance that is the largest of these 3 angles. So its x-coordinate needs to be sqrt(3)/2. So we write
that there. For Pi/4, for that angle, the x distance is a little smaller. So its the middle size number: sqrt(2)/2. Then for the angle Pi/3, the x distance is the smallest of the three. So it has to be 1/2 for the x-coordinate. Alright, now lets fill in the y values for those points. Its the exact same idea. For the angle Pi/6, the y distance, the height, is the lowest of those angles. So it needs to be 1/2. The smallest value. At Pi/4, the height, the y distance, is the middle size. So it needs to be sqrt(2)/2. And then at Pi/3, the y distance, the height, is the largest of the angles. So it needs to be sqrt(3)/2. Alright, now you can figure out all the other points for the other angles in the circle. Its the exact same idea. The only thing thats different is that you might have a negative sign. The only thing you need to watch out for is maybe having a negative sign for your coordinates. Like here. When you were in the first quadrant the first quarter of the circle, everything was positive. All of these coordinates were positive numbers. But if youre in the second quadrant, the second quarter, youve actually moved in the negative x direction. Left. So of all the x numbers will be negative.. have a negative sign. But the ys are still positive, because youre in the positive half of the y-axis. Then in the third quadrant, the third quarter youre in the negative x direction and the negative y direction because youve moved left and down. So both the xs and ys will have negative signs here. In the fourth quadrant, the fourth quarter youre in the positive
because you moved right and youre in the negative y direction, because you moved down. So positive x-coordinate. Negative y-coordinate. So I just want to remind you for all of these points, all of these angles the first number in the point is the cos value. And the second number is the sin. So here for 3Pi/4 the cos is -sqrt(2)/2. And the sin of 3Pi/4 is sqrt(2)/2. So if you have to figure out the cos of Pi/3 its the first value in the point. The answer is 1/2. If someone asks you “Whats the sign of 7Pi/6?” Go to your unit circle. sin is the second value in the point, so the sin of 7Pi/6 is just -1/2. That is the answer. Alright, just a few reminders for you. This whole time we were working in radian angles. All these angles, Pi/6, those are all radian angles. You may need to work in degrees. And all of them have a corresponding degree angle. You might already know that Pi is 180 degrees. So that would mean half of that, Pi/2, is 90 degrees. This of course is 0 degrees. And that would make Pi/6, 30 degrees. Pi/4, 45 degrees. Pi/3, 60 degrees. 90 degrees and so on. Also, if youre trying to memorize the unit circle I know its a lot to take in. Its kind of a mess. But just remember the patterns, because those patterns are what will make it a lot faster to learn and a lot easier to learn. So I hope that helped you figure out the unit circle. Unit circles are.. lots.. of.. fun. Its OK. You dont have to like math. but you can like my video.. so if you did, please click Like or subscribe!
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