Unit 7, Lesson 4
The Unit Circle (Part 2)
Algebra 2
4.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
Here is a circle with a radius of 1 and with some radii drawn.
A circle with center at the origin of an x y plane. The circle has a diameter of 20 units. Horizontal dashed lines are drawn 10 units above the origin and 10 units below the origin. Vertical dashed lines are drawn 10 units to the right of the origin and 10 units to the left of the origin. A segment is drawn from the origin to a point on the circle in the first quadrant and is labeled pi over 3. A segment is drawn from the origin to a point on the circle in the second quadrant and is labeled 3 pi over 4. A segment is drawn from the origin to a point on the circle in the third quadrant and is labeled 13 pi over 12. A segment is drawn from the origin to a point on the circle in the fourth quadrant and is labeled 5 pi over 3.
Draw and label these angles with the positive -axis as the starting ray for each angle, and moving in the counterclockwise direction. Four of these angles, one in each quadrant, have been drawn for you. Be prepared to share any strategies that you used to make the angles.
Given any point in a quadrant on the unit circle and given its associated angle, like shown here, we can make some statements about other points that must also be on the unit circle.
A circle with center at the origin of an x y plane. Point R lies on the outside of the curve, in the third quadrant, closer to the x axis than the y axis. The angle from the x axis, in the first quadrant, to point R is labeled a.
For example, if the coordinates of are and is radians, then there is a point, , in Quadrant I with coordinates . Since is -radian from a half circle, the angle associated with point must be -radian. Similarly, there is a point at with an angle -radian greater than a half circle. This means point is at angle -radian, since .
What is the matching point to in Quadrant IV? (A point at and angle radians.)
In future lessons, we’ll learn about how to find the coordinates of point ourselves using its angle, , and what we know about right triangles.