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The Unit Circle PreCalculus Right now… Get a protractor, 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, 1) (-1,0) |-------1-------| (0 , 0) (1,0) (0, -1) ** Note – You should be writing this information on the white paper! Using what you learned in 4.1 about sketching angles, what are the angle measurements of each of the four angles we just found? 90° π/2 0° 0 360° 2π 180° π 270° 3π/2 Using the Blue Circle Fold the circle in half twice. You should now be holding something that looks like a quarter of a pie. Hold the piece with the two folds on the left and the single fold on the bottom. Using the Blue Circle MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE BOTTOM. With your protractor in the corner of the pie piece, draw a 45° angle. Start cutting here, then over to the line Hold the corner of the pie piece and cut along the line you just drew (cut slightly above the corner, not through it). Once you reach the outside of the circle, cut down to the single fold, forming a 45-4590 right triangle. 45-45-90 Blue Triangle We know that a 45-45-90 triangle has side lengths: 2 1 1 But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle). So the new side lengths of the 45-45-90 triangle are: 1 2 2 2 2 45-45-90 Blue Triangle Holding the triangle with the single fold down and double fold to the left, label each side on the triangle. Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I. 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. 135° 3π/4 45° π/4 225° 5π/4 315° 7π/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. ( 2, 2 2 2 3π/4 135° ) 2 2 ( , 2 45° π/4 2 ) 2 2 2 2 5π/4 ( 2 2 , 225° ) 2 2 7π/4 315° ( 2 , 2 2) 2 Using the Green Circle Fold the circle in half twice. You should now be holding something that looks like a quarter of a pie. Hold the piece with the two folds on the left and the single fold on the bottom. Using the Green Circle MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE BOTTOM. With your protractor in the corner of the pie piece, draw a 60° angle. Start cutting here, then over to the line Hold the corner of the pie piece and cut along the line you just drew (cut slightly above the corner, not through it). Once you reach the outside of the circle, cut down to the single fold, forming a 30-6090 right triangle with the 60° at the bottom. 30-60-90 Green Triangle We know that a 30-60-90 triangle has side lengths: 2 3 60° 1 But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle. 1 So the new side lengths of the 30-60-90 triangle are: 3 2 60° 1 2 30-60-90 Green Triangle Holding the triangle with the single fold down and double fold to the left, label each side on the triangle. Unfold the triangle (so it looks like a butterfly) and 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. 120° 2π/3 60° π/3 5π/3 4π/3 240° 300° 60° Reference Angles - Coordinates Use the side lengths we labeled on the QI triangle to determine coordinates. 2π/3 ( 1 , 3 2 π/3 60° 120° ) 2 ( 1 3 , 2 2 ) 3 2 1 2 5π/3 4π/3 ( 1 2 , 3 2 ) 240° 300° ( 1 2 , 3) 2 Using the Yellow Circle Fold the circle in half twice. You should now be holding something that looks like a quarter of a pie. Hold the piece with the two folds on the left and the single fold on the bottom. Using the Yellow Circle MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE BOTTOM. With your protractor in the corner of the pie piece, draw a 30° angle. Start cutting here, then over to the line Hold the corner of the pie piece and cut along the line you just drew (cut slightly above the corner, not through it). Once you reach the outside of the circle, cut down to the single fold, forming a 30-6090 right triangle with the 30° at the bottom. 30-60-90 Yellow Triangle We know that a 30-60-90 triangle has side lengths: 2 30° 1 3 But… Our right triangle has a hypotenuse of 1 (because that’s the radius of the circle. So the new side lengths of the 30-60-90 triangle are: 1 1 2 30° 3 2 30-60-90 Yellow Triangle Holding the triangle with the single fold down and double fold to the left, label each side on the triangle. Unfold the triangle (so it looks like a butterfly) and 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. 150° π/6 30° 5π/6 210° 7π/6 11π/6 330° 30° Reference Angles - Coordinates Use the side lengths we labeled on the QI triangle to determine coordinates. ( 3, 1 2 ) 3 ( 2 2 150° 5π/6 , 1 2 ) 30° π/6 1 2 3 2 ( 3 2 7π/6 210° 330° 1 ( , 2 ) 3 2 11π/6 , 1 2 ) Final Product The Unit Circle