Unit Circle Construction Activity

Part One: Create the unit circle with Val

1. Graph the equation for the unit circle: 2. Click on the GEOMETRY button. 3. Click on the POINT tool. Add three points: One point at the origin (0,0) Another point at (1,0) A third point anywhere on the circle (This point is VAL) 4. Click on the SEGMENT tool. Draw the radius from the center (0,0) to the point on the circle (VAL). 5. Click on the ANGLE tool. Create the standard position angle by selecting the three points in order (1,0) then (0,0) then VAL 6. Click on the CALCULATOR button. Rename the point on the circle as VAL. 7. Click on the three dots next at the end of the line. Click on SHOW LABEL. Drag VAL around the circle. Her standard position angle should change as she travels around the circle.

Part Two: Val's Height

1. Drag VAL into quadrant I 2. In the INPUT line, type: X=(x(VAL),0). A point should appear on the x-axis, directly under VAL. 3. In the INPUT line, type: h=segment(VAL,X). 4. Select the input line for h. Click on the three dots at the end of the line. Under SHOW LABEL, select NAME & VALUE. 4. Drag Val. Her angle and her height should change as you drag her around the unit circle.

At what angles will Val be 1/2 foot above the x-axis?

How high is Val at 75˚? At what other angles will she be at this height (above or below)?

How high is Val at 110˚? At what other angles will she be at this height?

Part 3: Making Reference Triangles

1. Drag VAL back to quadrant I. 2. Click on the Geometry Button. 3. Click on the POLYGON Button. Click on the points in this order: VAL, Origin, X, VAL A shaded right triangle should be formed. 5. Click on the REFLECT ABOUT LINE button. Click in this order: VAL, then the shaded triangle, then the y-axis. A reflection of the triangle should appear in Quadrant II. 6. Now click in this order: VAL, then shaded triangle, then x-axis. A reflection of the triangle should appear in Quadrant IV. 7. Create the mirror-image in Quadrant III (you figure out how!) You should now have 4 congruent triangles, all mirror-images across the respective axes. If you drag VAL, you should see all four triangles change too.

Drag VAL to an angle of 40˚. What are the standard position angles for the other three points?

Drag VAL to an angle of 160˚. What are the standard position angles for the other three points?

Drag VAL to an angle of 200˚. What are the standard position angles for the other three points?

Drag VAL to an angle of 270˚. What happens to the triangles? Why? At what other angles will this happen?

Drag VAL to an angle of 305˚. What are the standard position angles for the other three points?

Suppose you know Val's angle in Quadrant I. How could you find the other three angles?

Suppose you know Val's angle in Quadrant II. How could you find quadrant I standard position angle?

Suppose you know Val's angle in Quadrant III. How could you find the quadrant I standard position angle?

Suppose you know Val's angle in Quadrant IV. How could you find the quadrant I standard position angle?