What Is the Unit Circle?

The unit circle is a circle of radius 1 centred at the origin (0, 0) on the Cartesian plane. Its equation is x² + y² = 1. Every point on the unit circle can be written as (cos θ, sin θ), where θ is the angle measured anticlockwise from the positive x-axis.

The Unit Circle with Key Points

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Q1

Q2

Q3

Q4

Point P(cos θ, sin θ) lies on the unit circle at angle θ

The ASTC Rule (All Stations To Central)

This mnemonic tells us which trigonometric ratios are positive in each quadrant:

Q1: All positive

Q2: Sin positive

Q3: Tan positive

Q4: Cos positive

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It allows us to find trigonometric values for any angle by relating it back to the first quadrant.

Quadrant	Angle Range	Reference Angle
Q1	0° < θ < 90°	θ
Q2	90° < θ < 180°	180° − θ
Q3	180° < θ < 270°	θ − 180°
Q4	270° < θ < 360°	360° − θ

For example, sin(150°): the reference angle is 180° − 150° = 30°. Since 150° is in Q2 where sine is positive, sin(150°) = sin(30°) = 1/2.

Key Coordinates on the Unit Circle

By using the special triangles (30-60-90 and 45-45-90), we can determine exact coordinates for key points on the unit circle:

Angle	Radians	cos θ	sin θ	Point (x, y)
0°	0	1	0	(1, 0)
30°	π/6	√3/2	1/2	(√3/2, 1/2)
45°	π/4	√2/2	√2/2	(√2/2, √2/2)
60°	π/3	1/2	√3/2	(1/2, √3/2)
90°	π/2	0	1	(0, 1)

Key Vocabulary

Unit Circle

A circle with radius 1 centred at the origin. Its equation is x² + y² = 1.

Reference Angle

The acute angle between the terminal side of an angle and the x-axis.

Terminal Side

The ray that rotates from the initial side to form an angle at the origin.

Quadrant

One of the four regions of the Cartesian plane, determined by the signs of x and y.

Worked Examples

1

Find the exact value of cos(120°).

Step 1: 120° is in Quadrant 2. The reference angle is 180° − 120° = 60°.

Step 2: In Q2, cosine is negative (ASTC rule).

Answer: cos(120°) = −cos(60°) = −1/2

2

Find the exact value of sin(225°).

Step 1: 225° is in Quadrant 3. The reference angle is 225° − 180° = 45°.

Step 2: In Q3, sine is negative.

Answer: sin(225°) = −sin(45°) = −√2/2

3

Find the exact coordinates of the point on the unit circle at 5π/6.

Step 1: 5π/6 = 150°, which is in Q2. Reference angle = 180° − 150° = 30°.

Step 2: In Q2, cos is negative and sin is positive.

Answer: The point is (−√3/2, 1/2)

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

On the unit circle, what does the x-coordinate of a point represent?

sin θ tan θ cos θ The radius

Question 2

What is the exact value of sin(330°)?

1/2 −√3/2 √3/2 −1/2

Question 3

In which quadrant is tan θ positive and cos θ negative?

Quadrant 3 Quadrant 2 Quadrant 1 Quadrant 4

Question 4

What is the reference angle for 210°?

45° 30° 60° 150°

Question 5

What is the exact value of cos(5π/4)?

√2/2 1/2 −√2/2 −1/2

The Unit Circle