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The Unit Circle

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The Unit Circle

PreCalculus

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!

Definition: A circle centered at the origin with a radius of exactly one unit.

(0, 1)

|-------1-------|

(-1,0)

(0 , 0)

(1,0)

** Note – You should be writing this information on the white paper!

(0, -1)

π/2

90°

0°

0

2π

180°

360°

π

270°

3π/2

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.

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.

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 45-45-90 right triangle.

Start cutting here, then over to the line

We know that a 45-45-90 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 45-45-90 triangle are:

1

1

1

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.

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°

45°

π/4

3π/4

225°

315°

7π/4

5π/4

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.

( , )

3π/4

( , )

135°

45°

π/4

7π/4

5π/4

225°

315°

( , )

( , )

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.

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.

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 30-60-90 right triangle with the 60° at the bottom.

Start cutting here, then over to the line

We know that a 30-60-90 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 30-60-90 triangle are:

60°

1

60°

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.

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°

π/3

2π/3

5π/3

4π/3

240°

300°

Use the side lengths we labeled on the QI triangle to determine coordinates.

2π/3

π/3

60°

120°

( , )

( , )

5π/3

4π/3

( , )

( , )

240°

300°

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.

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.

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 30-60-90 right triangle with the 30° at the bottom.

Start cutting here, then over to the line

We know that a 30-60-90 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 30-60-90 triangle are:

30°

1

30°

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.

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.

π/6

150°

30°

5π/6

11π/6

330°

210°

7π/6

Use the side lengths we labeled on the QI triangle to determine coordinates.

( , )

( , )

150°

30°

π/6

5π/6

7π/6

11π/6

330°

210°

( , )

( , )