Unwrapping the Unit Circle

Essential Question: What are the graphs of the sine and cosine functions? Enduring Understanding: Know the characteristics of the sine graph and cosine graph.

Materials needed: Paper Compass Protractor Ruler Markers Pencil

Fold your paper lengthwise. Then open.

Open your compass to a radius of cm cm

With your pencil point on the tick mark and the compass point on the fold line, draw a circle. Use your pencil to mark a dot in the center of the circle.

Using your ruler draw both a horizontal axis and a vertical axis intersecting at the center of the circle. Extend both axes 1-2 cm beyond the edge of the circle.

Since this is a unit circle, make a tick mark at each intersection point of the axes and the circle. Then label the tick marks accordingly. 1 1

1 1 Using your protractor, make a small tick mark every 15° from 0° to 360° around the circle. Label the tick marks on the outside of the circle. 0° 15° 30° 45°

1 1 Approximately 2 cm from the right side of the horizontal axis, draw a 48 cm horizontal line on the fold line. On the left side of this line, draw a vertical line at least the same length as the vertical axis on the circle.

1 On the new vertical axis, draw a tick mark for 1 and -1 directly across from those on the vertical axis of the circle. Label similarly. 1

1 1 0° 15° 30° 45° 60° ° Beginning at the intersection point of the new axes, draw a tick mark every 2 cm and label in 15°-increments from 0° to 360°. 1

The graph of the sine function. Remember that on a unit circle, the sine of the angle is the vertical of the reference triangle

1 1 0° 15° 30° 45° 60° ° For each angle on the circle, using your ruler measure the vertical length from the horizontal axis to the point corresponding to that angle. 1 0° 15° 30° 45° Then draw a line segment the same length above the corresponding tick mark in the coordinate plane on the right.

1 1 0° 15° 30° 45° 60° ° Continue this process for each angle of the circle from 0° to 360°. 1 0° 15° 30° 45° Notice that after 180° the vertical segment is below the horizontal axis indicating a negative value. The segment should be drawn similarly in the coordinate plane on the right after 180° as well.

1 1 0° 15° 30° 45° 60° ° After you have completed the process from 0° to 360°, use your pencil to “smoothly” connect the dots at the far ends of the segments 1

1 -360° -270° -180° -90° 0° 90° 180° 270° 360° -2  -3  /2 -  -  /2 0  /2  3  /2 2  Sketch the Sine Graph below. Domain: Range:

The graph of the cosine function. Remember that on a unit circle, the cosine of the angle is the horizontal of the reference triangle

1 1 0° 15° 30° 45° 60° ° For each angle on the circle, using your ruler measure the horizontal length from the vertical axis to the point corresponding to that angle. 1 0° 15° 30° 45° Then draw a vertical line segment the same length above the corresponding tick mark in the coordinate plane on the right.

1 1 0° 15° 30° 45° 60° ° Continue this process for each angle of the circle from 0° to 360°. 1 0° 15° 30° 45° Notice that after 90° to 270° the segments are to the left of the vertical axis. These segments represent negative values so the segments should be drawn below the x-axis in the coordinate plane on the right.

1 1 0° 15° 30° 45° 60° ° After you have completed the process from 0° 1o 360°, use your pencil to “smoothly” connect the dots at the far ends of the segments 1

1 -360° -270° -180° -90° 0° 90° 180° 270° 360° -2  -3  /2 -  -  /2  /2  3  /2 2  Sketch the Cosine Graph below. Domain: Range: