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Unit Circle: How to Graph Sine and Cosine Functions


Easy Ways to Graph Sine and Cosine Functions

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Sine and Cosine Functions Transcript:

Let me draw a big unit circle here. That’s a circle of radius 1. Let’s remember where these angles are, 0, of course is on the positive x-axis. Here’s the x-axis, here’s the y-axis, there is zero, ? over 2, remember that’s the same as a 90-degree angle, that’s a right angle, so that’s up there ? over 2; ? is over here, that’s a 180 degrees; 3? over 2, is down here, and 2? is right here. We want to find the cosine and sine of each one of those angles. Now remember, cosine and sine, by definition, are the x and y coordinates of those angles. What are these x and y coordinates? The 0 angle, it’s x-coordinate is 1, and it’s y-coordinate is 0. That tells us that cos(0) is 1, and that sin(0) is 0. Pi over 2 is up here, and so it’s cosine is the x-coordinate, well, the x-coordinate of that point is 0. The y-coordinate is 1, and so that’s the sin of ? over 2. Pi is over here at (-1,0), so that’s the cosine and sine of ?. Cos(?) is -1, sin(?) is 0. Finally, 3? over 2 is down here at (0,-1), so that’s the cosine and sine of 3? over 2. Cos(0), sin(?/2), is -1. And one more, 2? is back in the same place as 0, so it has the same cosine and sine. Cos(2?) is 1, sin(2?) is 0. That’s how you figure out the cosines and sines of angles. As you graph them on the unit circle, and then you look at the x and y coordinates. The x-coordinate is always the cosine, and the y-coordinate is always the sine. By the way, these are very common values, 0, ?/2, 3?/2, and 2?. You should really know the sines and cosines of these angles by heart. They come up so often in trigonometry context that it’s worth memorizing these things, and being able to sort of regurgitate them very very quickly. If you ever forget them though, if you ever can’t quite remember what the cosine of ?/2, or the sine of 3?/2 is, Then, what you do is draw yourself a little unit circle, and you figure what the x and y coordinates are and you can always work them out. It’s worth memorizing them to know them quickly, but if you ever get confused, you are not quite sure, just draw yourself a unit circle and you’ll figure them out quickly. We’re going to use these values, so I hope you will remember these values for the next example.

About Dr. William Murray, Ph.D.

Graphing Secant and Cosecant Functions Trigonometry Educator

Most of us are unaware of how trigonometry factors into our daily lives. Dr. William Murray is very aware of the importance of trigonometry and he is ready to help you master it all. His in-depth course covers everything from Functions, Identities, and Complex/Polar Coordinates to Word Problems and the Law of Sines.  This course meets or exceeds all state standards and is essential to those having trouble with trigonometry in any setting. Professor Murray received his Ph.D from UC Berkeley, B.S. from Georgetown University, and has been teaching in the university setting for 10+ years.

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