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Suppose there is a point at on the unit circle.
For each tick mark on the horizontal axis, plot the value of , where is the measure of an angle in radians. Use the class display of the unit circle, the unit circle from an earlier lesson, or technology to estimate the value of .
A blank graph. Horizontal axis, theta, 0 to 2 pi, by pi over 12. Vertical axis, y, negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis.
For each tick mark on the horizontal axis, plot the value of . Use the class display of the unit circle, the unit circle from an earlier lesson, or technology to estimate the value of .
A blank graph. Horizontal axis, theta, 0 to 2 pi, by pi over 12. Vertical axis, y, negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis.
What do you notice about the two graphs?
Explain why any angle measure between 0 and gives a point on each graph.
Could these graphs represent functions? Explain your reasoning.
Using the unit circle, we can make sense of and for any angle measure between 0 and radians. For an angle , starting at the positive -axis, there is a point, , where the terminal ray of the angle intersects the unit circle. The coordinates of that point are .
A circle with center, point B, at the origin of an x y plane. Point A lies on the outside of the circle, on the x axis, to the right of the origin. Point C lies in the second quadrant on the outside of the circle. Angle A B C is labeled theta.
But what if we wanted to think about just the horizontal position of point as goes from 0 to ? The horizontal location is defined by the -coordinate, which is . If we graph , we get:
A graph. Horizontal axis, theta, scale 0 to 2 pi by pi over 8. Vertical axis, y, scale negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis. A curve is drawn and labeled y equals cosine theta.
This graph is 1 when is 0 because the -coordinate of the point at 0 radians on the unit circle is . The graph then decreases to -1 (the smallest -value on the unit circle) before increasing back to 1.
We can do the same for the -coordinate of a point on the unit circle by graphing :
A graph. Horizontal axis, theta, scale 0 to 2 pi by pi over 8. Vertical axis, y, scale negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis. A curve is drawn and labeled y equals sine theta.
This graph is 0 when is 0, increases to 1 (the greatest -value on the unit circle), then decreases to -1 before returning to 0.