I understand that the period for sine and cosine functions are theta (in radians) + 2pi.

But if you have a point on the unit circle in the first quadrant, and you drew a line parallel to the x-axis until you reached the unit circle in quadrant 2, wouldn't that be the same y-value, thus the same value for sin theta?

Yes. It would be the same value.

If you repeated the process from the point in quadrant 1 and drew a line parallel to the y-axis until you reached the unit circle in quadrant 4, wouldn't that be the same x-value, thus the same value for cos theta?

Yes. It would be the same value.

I don't understand how this works, if the period is 2pi, then why are there other points on the unit circle that would give the same value for x and y, therefore sine and cosine respectively?

If you take a graph of sin(x) or cos(x) plotted, say, from x = -4·Pi to x = 4·Pi, and then drew a horizontal line through it, say y=1/2, you'll see that this line intersects the graph at two locations, within any interval of width 2·Pi.

Another way of thinking about it is with respect to the symmetry of a circle. As the terminal ray of the angle (in standard position with a unit circle) rotates about its vertex, the point of its intersection with the circle is traveling around the circumference in a counterclockwise direction. For y=sin(x) as angle x increases from x=0 to x=Pi/2 radians, this point is rising in Quadrant I (the y-coordinate is increasing from 0 toward 1).

Once angle x has increased past Pi/2, the point moves past the point (0,1) and starts descending in Quadrant II toward the point (-1,0). Due to the symmetry of the circle, the point's y-coordinate will pass through every value on its way back to zero that it just finished passing on its way up in the first quadrant. That is, sin(x) takes on every value between 0 and 1 twice, as the terminal ray rotates through Quadrant I and Quadrant II.

I think the best thing for you to do at this point, is to google keywords

sine cosine unit circle animation, and check out a lot of the animated .GIFs, videos, interactive geometry or computer algebra sites, and watch all the different graphical representations of the periodicity of the sine and cosine functions.

After that, if you still have questions about why sin(x) and cos(x) values repeat within one period, let us know. :cool:

Here's one short video showing sin(x), but do check out a lot more, by googling.

[video=youtube;Ohp6Okk_tww]https://www.youtube.com/watch?v=Ohp6Okk_tww[/video]