# Unit Circle: Definition, Coordinates, Chart, Examples

## TL;DR

The unit circle is the circle of radius 1 centred at the origin, where every point on the rim has coordinates (cos θ, sin θ) for the angle θ measured from the positive x-axis. This article covers the definition, the equation x² + y² = 1, why the coordinates are cosine and sine, the special angles in both degrees and radians, six worked examples, and the common mistakes.

## What Is the Unit Circle?

The **unit circle** is the circle of **radius 1** centred at the **origin** (0,0) of the coordinate plane. Because the radius is exactly 1, its equation comes straight from the Pythagorean theorem applied to any point (x,y) on the rim:

x² + y² = 1.

Now place a point P on the circle by rotating a radius counter-clockwise from the positive x-axis through an angle θ. The coordinates of P are exactly the cosine and sine of that angle:

P = (cos θ, sin θ).

That is the whole power of the unit circle: the x-coordinate of the point **is** cos θ, and the y-coordinate **is** sin θ. No ratios to compute, no triangle to draw, the position on the circle hands you both values directly. This material appears in NCERT Class 11, Chapter 3 (Trigonometric Functions) and under CCSS-M HSF-TF.A.2, which extends the trig functions to all real angles using the unit circle.

## Why the Coordinates Are Cosine and Sine

This is worth deriving once, because it is the bridge between the triangle definition and the circle. Take the point P at angle θ in the first quadrant and drop a perpendicular from P to the x-axis. This builds a right triangle whose hypotenuse is the radius (length 1), whose horizontal leg is the x-coordinate of P, and whose vertical leg is the y-coordinate of P.

From the right-triangle definitions, with the hypotenuse equal to 1:

cos θ = adjacent/hypotenuse = x/1 = x, sin θ = opposite/hypotenuse = y/1 = y.

So x = cos θ and y = sin θ, the coordinates _are_ the trig values, because dividing by a hypotenuse of 1 leaves them unchanged. This double-anchoring matters: the same sin 30° = 1/2 is the _ratio_ opposite-over-hypotenuse in a triangle and the _y-coordinate_ of the point at 30° on the circle. The unit circle then carries the definition past 90°, where a triangle can no longer fit, by simply letting the point travel into the other quadrants. And because x² + y² = 1, substituting gives the most-used identity in trigonometry, cos² θ + sin² θ = 1.

## What Is the Tangent on the Unit Circle?

Tangent is the third value, and it comes from the other two:

tan θ = sin θ/cos θ = y/x.

On the unit circle, the tangent of an angle is the y-coordinate divided by the x-coordinate of the point, the slope of the radius drawn to P. This is why tan 90° is undefined: at 90° the point is (0,1), so x=0 and dividing by zero has no value.

## The Special Angles and Their Coordinates

A handful of angles appear so often that their coordinates are worth knowing on sight. In the first quadrant, the special angles and their (cos θ, sin θ) coordinates are:

| Angle (degrees) | Angle (radians) | cos θ (x) | sin θ (y) | tan θ |
| --- | --- | --- | --- | --- |
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |

### How signs change by quadrant

As P travels round, the signs of cosine and sine follow the signs of the x- and y-coordinates: both positive in Quadrant I, cosine negative in Quadrant II, both negative in Quadrant III, sine negative in Quadrant IV. (The mnemonic "All Students Take Calculus" names which function stays positive in each quadrant.)

## Examples of the Unit Circle

With the coordinate meaning and the special angles in hand, here are values being read off the circle. The problems move from a direct coordinate read to using the circle past 90° and converting units.

### Example 1 - Use the unit circle to find cos 60° and sin 60°

The point at 60° has coordinates (1/2, √3/2), and the coordinates _are_ cosine and sine:

cos 60° = 1/2, sin 60° = √3/2.

Final answer: cos 60° = 1/2, sin 60° = √3/2.

### Example 2 - Find cos 90° and sin 90° from the unit circle

_Correct._ At 90° the point sits at the top of the circle, (0,1), so:

cos 90° = 0, sin 90° = 1.

Final answer: cos 90° = 0, sin 90° = 1.

### Example 3 - Find the coordinates of the point at 180° on the unit circle

180° points along the negative x-axis, so the point is the left-hand intercept:

(cos 180°, sin 180°) = (-1,0).

Final answer: (-1,0).

### Example 4 - Convert 45° to radians and give the coordinates at that angle

Convert: 45° × π/180° = π/4. The point at 45° lies on the line y=x in Quadrant I:

(cos π/4, sin π/4) = (√2/2, √2/2).

Final answer: π/4 radians, coordinates (√2/2, √2/2).

### Example 5 - Find tan 30° using the unit circle coordinates

At 30° the point is (√3/2, 1/2), and tangent is y over x:

tan 30° = sin 30°/cos 30° = 1/2/√3/2 = 1/√3.

Final answer: tan 30° = 1/√3.

### Example 6 - The point at angle θ on the unit circle is (−√2/2, √2/2). Find θ

Cosine is negative and sine positive, so the point is in Quadrant II. The reference values match 45°, and the Quadrant II angle with that reference is:

θ = 180° - 45° = 135° = 3π/4.

Final answer: θ = 135°.

## Why the Unit Circle Matters

The unit circle is not a memorisation chart, it is the move that turns trigonometry from "ratios in a triangle" into a tool that works for any angle, including the rotating and repeating kind.

- **It defines sine and cosine for every angle.** A right triangle can only hold angles below 90°. The unit circle extends sin and cos to 180°, 270°, full turns, and negative angles, by letting the point keep travelling round.
- **It is where the waves come from.** Track the y-coordinate of P as it spins steadily and you trace a sine wave; the x-coordinate traces a cosine wave. Every oscillation modelled with sin or cos, sound, alternating current, tides, light, is a shadow of a point going round this circle.
- **It powers rotation and navigation.** Rotating any object in a video game, a robot arm, or a satellite's orientation uses (cos θ,sin θ) to place a rotated point. GPS and computer graphics lean on exactly these coordinates.
- **It makes the Pythagorean identity obvious.** Because x²+y²=1, the identity cos² θ + sin² θ = 1 is not something to memorise, it is the circle's own equation rewritten.

## Common Mistakes

### Mistake 1: Swapping the x- and y-coordinates for cosine and sine

**Correct way:** The point is always **(cos θ, sin θ)**, x first. At 30° the point is (√3/2, 1/2).

### Mistake 2: Forcing a right triangle for angles of 90° or more

**Correct way:** Read the coordinate of the point on the circle. The reason for the unit circle is to define functions where a triangle cannot.

### Mistake 3: Mixing degrees and radians without converting

**Correct way:** Use 180° = π radians to convert before reading the circle.

## Key Takeaways

- The **unit circle** is the circle of radius 1 centred at the origin, with equation x² + y² = 1.
- Every point on it is (cos θ, sin θ), so cosine is the x-coordinate and sine is the y-coordinate.
- The special angles 30°, 45°, 60° (π/6, π/4, π/3) and the quadrant points are worth knowing.
- The most common mistake is swapping x and y; the point is always (cos θ, sin θ).
- The unit circle extends sine and cosine to every angle and is where sine and cosine waves come from.