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Amy Ferguson Moncure

IB AA · Unit 2 · 18A Radian Measure

Points on the Unit Circle

Reduce the angle to one turn, name where the ray lands, and read the coordinates from the reference angle.

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Key Ideas Easy Medium Hard

x = cos θ, y = sin θ

One full turn is 2π

What to Remember

Coterminal angles

The point depends on the ray, not on how many extra turns you took.

Add or subtract 2π until the angle is from 0 to 2π.

Coordinates

On the unit circle, x = cos θ and y = sin θ.

π/6 → (square root of 32, 12)

π/4 → (square root of 22, square root of 22)

π/3 → (12, square root of 32)

Other quadrants keep those absolute values and change the signs.

Easy

One turn, positive

The angle is already between 0 and 2π.

Example
Mark the point on the unit circle for π4. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.

π4 is already from 0 to 2π.

π4 is on Quadrant I.

The reference angle is π4.

Use that reference point, then apply the signs for this quadrant.

The point is (square root of 22, square root of 22).

Answer: Quadrant I, (square root of 22, square root of 22)

1
Mark the point on the unit circle for π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

π6 is already from 0 to 2π.

π6 is on Quadrant I.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (square root of 32, 12).

Answer: Quadrant I, (square root of 32, 12)

2
Mark the point on the unit circle for π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

π3 is already from 0 to 2π.

π3 is on Quadrant I.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (12, square root of 32).

Answer: Quadrant I, (12, square root of 32)

3
Mark the point on the unit circle for π2. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

π2 is already from 0 to 2π.

π2 is on the positive y-axis.

The point is (0, 1).

Answer: the positive y-axis, (0, 1)

4
Mark the point on the unit circle for 4π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

4π3 is already from 0 to 2π.

4π3 is on Quadrant III.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (−12, −square root of 32).

Answer: Quadrant III, (−12, −square root of 32)

5
Mark the point on the unit circle for 11π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

11π6 is already from 0 to 2π.

11π6 is on Quadrant IV.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (square root of 32, −12).

Answer: Quadrant IV, (square root of 32, −12)

Medium

Negative angles

Add 2π until the angle is positive.

Example
Mark the point on the unit circle for −5π4. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.

Add or subtract 2π until the angle is from 0 to 2π.

−5π4 + 2π = 3π4

3π4 is on Quadrant II.

The reference angle is π4.

Use that reference point, then apply the signs for this quadrant.

The point is (−square root of 22, square root of 22).

Answer: Quadrant II, (−square root of 22, square root of 22)

1
Mark the point on the unit circle for −π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−π6 + 2π = 11π6

11π6 is on Quadrant IV.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (square root of 32, −12).

Answer: Quadrant IV, (square root of 32, −12)

2
Mark the point on the unit circle for −2π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−2π3 + 2π = 4π3

4π3 is on Quadrant III.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (−12, −square root of 32).

Answer: Quadrant III, (−12, −square root of 32)

3
Mark the point on the unit circle for −π. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−π + 2π = π

π is on the negative x-axis.

The point is (−1, 0).

Answer: the negative x-axis, (−1, 0)

4
Mark the point on the unit circle for −5π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−5π6 + 2π = 7π6

7π6 is on Quadrant III.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (−square root of 32, −12).

Answer: Quadrant III, (−square root of 32, −12)

5
Mark the point on the unit circle for −7π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−7π6 + 2π = 5π6

5π6 is on Quadrant II.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (−square root of 32, 12).

Answer: Quadrant II, (−square root of 32, 12)

Hard

More than one turn

Subtract 2π, or add it, until only one turn is left.

Example
Mark the point on the unit circle for 11π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.

Add or subtract 2π until the angle is from 0 to 2π.

11π3 − 2π = 5π3

5π3 is on Quadrant IV.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (12, −square root of 32).

Answer: Quadrant IV, (12, −square root of 32)

1
Mark the point on the unit circle for 7π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

7π3 − 2π = π3

π3 is on Quadrant I.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (12, square root of 32).

Answer: Quadrant I, (12, square root of 32)

2
Mark the point on the unit circle for 13π4. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

13π4 − 2π = 5π4

5π4 is on Quadrant III.

The reference angle is π4.

Use that reference point, then apply the signs for this quadrant.

The point is (−square root of 22, −square root of 22).

Answer: Quadrant III, (−square root of 22, −square root of 22)

3
Mark the point on the unit circle for 17π6. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

17π6 − 2π = 5π6

5π6 is on Quadrant II.

The reference angle is π6.

Use that reference point, then apply the signs for this quadrant.

The point is (−square root of 32, 12).

Answer: Quadrant II, (−square root of 32, 12)

4
Mark the point on the unit circle for 7π2. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

7π2 − 2π = 3π2

3π2 is on the negative y-axis.

The point is (0, −1).

Answer: the negative y-axis, (0, −1)

5
Mark the point on the unit circle for −11π3. Give a coterminal angle from 0 to 2π, name the quadrant or axis, and give the coordinates.
Show Answer

Add or subtract 2π until the angle is from 0 to 2π.

−11π3 + 2π = −5π3

−5π3 + 2π = π3

π3 is on Quadrant I.

The reference angle is π3.

Use that reference point, then apply the signs for this quadrant.

The point is (12, square root of 32).

Answer: Quadrant I, (12, square root of 32)

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