| Start | We have already seen that the sine and cosine are lengths:
They can be viewed as lengths of a right triangle
whose angle is located at the origin
and another angle is on the unit circle.
And the sine is the height of this triangle, or if it is below the axis it is a negative value of the height. And, the cosine is this width here. We have already seen that. What I would like to talk about now is... how to visualize |
| 0:31 | the graph of the sine
and the cosine.
So here is that same circle again. I am interested in the sine. And here we see the green length here is the height. And, as we move a point around on the unit circle. So, we think of starting at 0, and that is here: 0. And, now, moving along the x axis I am going to put the angles-- first 0, Pi over 2 Remember we are doing everything in radians. |
| 1:00 | then
Pi which is a half turn.
3 Pi over 2 here, which is a three-quarter turn. And then, 2 Pi a full turn, back again to the beginning of the circle. And then, as we move this around notice that the height, just take the height here, and put it on the axis and we mark the point on the graph it corresponds to that height. So, as we grow from 0 to Pi/2 you can see here that the sine goes up and then it comes back down again after Pi/2. |
| 1:31 | Just as we are seeing here at this height.
Then it becomes negative. It shrinks to its minimum at -1, and then it comes back up again. We can do the same thing for cosine. Remember the cosine is the width of this triangle. Let me put the sine in so we can see the triangle. Cosine is the red line. And, no, that... Um, So, the width starts out being the full radius of the unit circle. |
| 2:01 | And then,
begins to shrink as we go to the
top of the projection on to the radius here
at 0. And, you see there that the
graph here it goes down to 0.
And, as we continue going, it continues to shrink because it is now behind the axis, so it is negative. And then, it begins to go back toward 0 and up again. So, we can draw the graphs just by thinking of this height or this width as we move along the axis. So, we have |
| 2:31 | the graph of the sine in green here.
The graph of the cosine in red here. You can see one looks very much like the other one. Just shifted a little bit. The sine goes through 0 because at the angle of 0 the height is 0. The cosine goes through 1 because at the angle of 0, the height is 1. And then, as we move along that sweeps out the graph. On this page which you can play with separately, we can do the same thing for the tangent. |
| 3:00 | Here, or for the other trig functions.
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