Unit Circle & Graphs of Sine and Cosine
This sketch tries to show the relationship between the measurements of a point on the unit circle and the definition of the sine and cosine functions.
Please spend time exploring and playing around on this applet.
Then please respond to the questions at the bottom.
1) What do the values along the x-axis represent? How did you come to this conclusion?
2) With "cosine" selected, move the cursor around the unit circle in the counter-clockwise direction. In addition to the coordinates of the point being displayed there is additional number that lags behind the point , but is also changing. What does this value represent? How do you know this/how were you convinced of your answer?
3) Still with "cosine" selected, continue to move the point around the unit circle. As you do so a segment is highlighted along the x-axis. The point leading this segment also has coordinates. What do these coordinates represent? If you are not entirely sure, finish step 4 and then return to this question.
4) With "cosine" selected and you have moved the point somewhere on the circle, stop. Write down the coordinates of the point on your circle: ( , ) and for the point tracing along the x-axis ( , ). Now deselect "cosine" and select "sine". Did either of your coordinate points change? If so, state which one and why the change happened. If not, describe why the coordinates did not change.
5) Finally with "sine" still selected return your circle coordinate point to (1,0). Now select "trace on" and move cursor once around the circle. Write your observations here:
6) Deselect "sine", return circle coordinate point to (1,0) and repeat step 5 with "cosine" selected. Write your observations here:
7) Can you explain why the cosine graph initially decreases and the sine value increases?
8) Why do sine and cosine not have the same number of x-intercepts?
9) What do the graphs have in common?
10) Can you explain why the range of both graphs is [-1,1]?