The Unit Circle

What Is the Unit Circle?

The unit circle is a circle with a radius of exactly 1, centered at the origin of a coordinate plane. While that sounds simple, it is arguably the single most important diagram in all of trigonometry. Every trigonometric function can be defined, visualized, and evaluated using this circle.

Why a radius of 1? Because when the hypotenuse of a right triangle equals 1, the definitions of sine and cosine reduce to pure coordinates. If you draw a radius from the origin to any point on the circle, the x-coordinate of that point equals the cosine of the angle, and the y-coordinate equals the sine of the angle. This elegant relationship is the foundation for everything that follows.

Formally, the unit circle is the set of all points (x, y) that satisfy the equation:

This equation is itself a direct consequence of the Pythagorean theorem. For any point on the circle, the horizontal distance x and vertical distance y form the legs of a right triangle whose hypotenuse is the radius, which is 1. Therefore x² + y² = 1² = 1.

The Unit Circle Diagram

Below is a complete diagram of the unit circle showing all the standard special angles in both degrees and radians, along with their exact coordinate values. Study this diagram carefully — it is the map you will refer to throughout your study of trigonometry.

Special Angles and Their Exact Values

The unit circle has 16 standard special angles that you should commit to memory. These angles are built from just three reference angles — 30°, 45°, and 60° — reflected into all four quadrants. Below is the complete table of exact values for sine, cosine, and tangent at every special angle.

How to Read Sin, Cos, and Tan from the Unit Circle

Reading trigonometric values from the unit circle is straightforward once you understand the coordinate relationship:

• Cosine (θ) = the x-coordinate of the point on the circle
• Sine (θ) = the y-coordinate of the point on the circle
• Tangent (θ) = sin(θ) / cos(θ) = y / x

For example, at 60° the point on the unit circle is (1/2, √3/2). Therefore cos(60°) = 1/2, sin(60°) = √3/2, and tan(60°) = (√3/2) / (1/2) = √3.

The remaining three trig functions are simply reciprocals:

Quadrant Signs — The ASTC Rule

Not every trig function is positive in every quadrant. The mnemonic ASTC (sometimes remembered as "All Students Take Calculus") tells you which functions are positive in each quadrant:

• Quadrant I (0°–90°): All functions are positive. Both x and y are positive, so sin, cos, and tan are all positive.
• Quadrant II (90°–180°): Only Sine is positive. The y-coordinate is positive but x is negative, so sin > 0 while cos < 0 and tan < 0.
• Quadrant III (180°–270°): Only Tangent is positive. Both x and y are negative, so sin and cos are both negative, but their ratio (tan) is positive.
• Quadrant IV (270°–360°): Only Cosine is positive. The x-coordinate is positive but y is negative, so cos > 0 while sin < 0 and tan < 0.

Reference Angles

A reference angle is the acute angle formed between the terminal side of your angle and the x-axis. It is always between 0° and 90°. Reference angles let you reduce any trig calculation to one of the three fundamental angles (30°, 45°, 60°) and then apply the correct sign using ASTC.

Here is how to find the reference angle α for an angle θ:

• Quadrant I: α = θ
• Quadrant II: α = 180° − θ
• Quadrant III: α = θ − 180°
• Quadrant IV: α = 360° − θ

Once you know the reference angle, evaluate the trig function at that reference angle (which will always be positive), then apply the appropriate sign from the ASTC rule.

Worked Examples

Memorization Strategies

Memorizing the entire unit circle may seem daunting, but there are several strategies that make it manageable:

Strategy 1: The Three Key Coordinates

You only need to remember three numbers: 1/2, √2/2, and √3/2. For the first-quadrant angles:

• At 30°: cos = √3/2, sin = 1/2 (cosine is the larger value)
• At 45°: cos = √2/2, sin = √2/2 (both are equal)
• At 60°: cos = 1/2, sin = √3/2 (sine is the larger value)

Notice the pattern: as the angle increases from 30° to 60°, the sine value increases and the cosine value decreases. The values simply swap between 30° and 60°.

Strategy 2: The Counting Pattern

Write the sine values for 0° through 90° using a counting trick:

The numerators under the radical simply count 0, 1, 2, 3, 4. Simplifying gives you 0, 1/2, √2/2, √3/2, 1. For cosine, the same pattern runs in reverse.

Strategy 3: Symmetry Across Quadrants

Once you know Quadrant I, all other quadrants are mirror images. The magnitudes are the same; only the signs change. Use the ASTC rule to assign signs. For instance, 150° has the same reference angle as 30°, so the coordinate magnitudes are identical — you just flip the sign of the x-coordinate because cosine is negative in Quadrant II.

Practice Problems