The Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is the bridge between right-triangle trig and the broader world of trig functions — allowing you to define sine and cosine for any angle, not just angles in a right triangle.

What Is the Unit Circle?

Draw a circle on the $xy$-plane with center $(0, 0)$ and radius 1. Pick any point on the circle and draw a line from the origin to that point. The angle $\theta$ is measured from the positive $x$-axis, going counterclockwise.

The key insight: the coordinates of that point are $(\cos\theta, \sin\theta)$.

• The $x$-coordinate is $\cos\theta$
• The $y$-coordinate is $\sin\theta$

This works because if you drop a vertical line from the point to the $x$-axis, you form a right triangle with hypotenuse 1 (the radius). Then $\cos\theta = \frac{\text{adjacent}}{1} = x$ and $\sin\theta = \frac{\text{opposite}}{1} = y$.

The Four Quadrants

As the angle $\theta$ increases from 0 to 360 degrees, the point moves counterclockwise around the circle, passing through four quadrants. The signs of sine and cosine change depending on which quadrant the angle is in.

The ASTC Rule (All Students Take Calculus)

A popular mnemonic for remembering the signs is All Students Take Calculus — or A S T C — starting in Quadrant I and going counterclockwise:

• A (Quadrant I) — All trig functions are positive
• S (Quadrant II) — only Sine is positive
• T (Quadrant III) — only Tangent is positive
• C (Quadrant IV) — only Cosine is positive

Since $\tan\theta = \frac{\sin\theta}{\cos\theta}$, tangent is positive whenever sine and cosine have the same sign (both positive in Q I, both negative in Q III).

Key Coordinates on the Unit Circle

These four points are the starting foundation:

At these angles, either sine or cosine is 0 and the other is $\pm 1$. You will learn the coordinates of the special angles (30°, 45°, 60°, etc.) in the Special Angles topic.

Connecting the Unit Circle to Right Triangles

For any angle $\theta$ in Quadrant I, imagine the right triangle formed by:

• The radius (from origin to the point on the circle) — this is the hypotenuse, with length 1
• The horizontal distance from the origin to the point — this is $\cos\theta$
• The vertical distance from the $x$-axis to the point — this is $\sin\theta$

Since the hypotenuse is 1:

$\sin\theta = \frac{\text{opposite}}{1} = \text{opposite} = y$

$\cos\theta = \frac{\text{adjacent}}{1} = \text{adjacent} = x$

This is why $\sin^2\theta + \cos^2\theta = 1$ — it is the Pythagorean theorem applied to a triangle with hypotenuse 1: $x^2 + y^2 = 1^2$.

Worked Examples

Example 1: Determining the Sign

What is the sign of $\sin(210°)$?

210 degrees is in Quadrant III (between 180° and 270°). In Q III, both sine and cosine are negative.

Answer: $\sin(210°)$ is negative.

Example 2: Finding Coordinates

A point on the unit circle is at angle $\theta = 60°$. We know that $\sin(60°) = \frac{\sqrt{3}}{2}$ and $\cos(60°) = \frac{1}{2}$. What are the coordinates?

The coordinates are $(\cos 60°, \sin 60°) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Answer: The point is at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \approx (0.5, 0.866)$.

Example 3: Using Symmetry

If $\cos(50°) = 0.6428$, what is $\cos(310°)$?

310° is in Quadrant IV. The reference angle is $360° - 310° = 50°$. In Q IV, cosine is positive.

$\cos(310°) = \cos(50°) = 0.6428$

Answer: $\cos(310°) = 0.6428$

Why the Unit Circle Matters

The unit circle extends trigonometry beyond right triangles. With right triangles, you can only handle acute angles (0° to 90°). The unit circle defines sine and cosine for any angle — 0° to 360° and beyond. This is essential for:

• AC circuits — electricians analyze alternating current using sine waves that cycle through all four quadrants
• Rotation and circular motion — describing positions on a wheel, gear, or orbit
• Graphing trig functions — the sine and cosine graphs are generated by tracing the unit circle

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The unit circle is a circle with radius 1 centered at the origin
• Any point on the unit circle has coordinates $(\cos\theta, \sin\theta)$
• ASTC tells you the signs: All (Q I), Sine (Q II), Tangent (Q III), Cosine (Q IV)
• The identity $\sin^2\theta + \cos^2\theta = 1$ comes directly from the Pythagorean theorem on the unit circle
• The unit circle extends trig beyond right triangles, defining sine and cosine for all angles

Return to Trigonometry for more topics in this section.