The Unit Circle

Right now… Get a scissors, and one copy of each circle (blue, green, yellow, white). Sit down and take everything BUT that stuff & your writing utensils(4 different colors if you have them) off your desk. Cut out the blue, green, and yellow circles. Put your name on the white paper. DO NOT cut the white circle!

The Unit Circle Definition: A circle centered at the origin with a radius of exactly one unit. | | (0, 0)(1,0)(-1,0) (0, 1) (0, -1) ** Note – You should be writing this information on the white paper!

Using what you learned about sketching angles, what are the angle measurements of each of the four angles we just found? 180° 90° 270° 0° 360° 2π2π π /2 π 3 π /2 0

Using the Blue Circle Draw a line all the way through the circle at the 45° angle and at 135° angle. Draw a line straight down to the x-axis from 45° and 135° forming a right triangle. Then draw a line straight up to the x-axis from the 225° and 315° forming a right triangle.

Blue Triangle We know that a triangle has side lengths: But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle). So the new side lengths of the triangle are: 1 1 1

Blue Triangle Now cut out your blue triangles and paste them on your white circle. This is what it should look like.

45° Reference Angles We know that the quadrant one angle formed by the triangle is 45°. That means each other triangle is showing a reference angle of 45°. What about in radians? Label the remaining three angles. 45°135° 315°225° π /4 5 π /4 3 π /4 7 π /4

45° Reference Angles - Coordinates Remember that the unit circle is overlayed on a coordinate plane (that’s how we got the original coordinates for the 90°, 180 °, etc.) Use the side lengths we labeled on the QI triangle to determine coordinates. 45°135° 315°225° (, ) π /4 3 π /4 5 π /4 7 π /4

Using the Green Circle Draw a line all the way through the circle at the 60° angle and at 120° angle. Draw a line straight down to the x-axis from 60° and 120° forming a right triangle. Then draw a line straight up to the x-axis from the 240° and 300° forming a right triangle.

Green Triangle We know that a triangle has side lengths: But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle. So the new side lengths of the triangle are: 1 60°

Green Triangle Glue it to the white circle with the triangle you just labeled in quadrant I, on top of the blue butterfly.

60° Reference Angles We know that the quadrant one angle formed by the triangle is 60°. That means each other triangle is showing a reference angle of 60°. What about in radians? Label the remaining three angles. 60° 120° 300°240° π /3 5 π /3 4 π /3 2 π /3

60° Reference Angles - Coordinates Use the side lengths we labeled on the QI triangle to determine coordinates. 60° 120° 300°240° (, ) π /3 2 π /3 4 π /3 5 π /3

Using the Yellow Circle Draw a line all the way through the circle at the 30° angle and at 150° angle. Draw a line straight down to the x-axis from 30° and 150° forming a right triangle. Then draw a line straight up to the x-axis from the 210° and 330° forming a right triangle.

Yellow Triangle We know that a triangle has side lengths: But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle. So the new side lengths of the triangle are: 1 30°

Yellow Triangle Glue it to the white circle with the triangle you just labeled in quadrant I, on top of the green butterfly.

30° Reference Angles We know that the quadrant one angle formed by the triangle is 30°. That means each other triangle is showing a reference angle of 30°. What about in radians? Label the remaining three angles. 30°150° 330° 210° π /6 7 π /6 5 π /6 11 π /6

30° Reference Angles - Coordinates Use the side lengths we labeled on the QI triangle to determine coordinates. 30° 150° 330° 210° (, ) π /6 7 π /6 5 π /6 11 π /6

Final Product

The Unit Circle