Unit Circle: Chart, Exact Values And How To Use It

What Is The Unit Circle?

The unit circle is a circle with a radius of exactly 1, centered at the origin of the coordinate plane. Every point (x, y) on it satisfies the equation x² + y² = 1.

That equation comes straight from the Pythagorean theorem. Draw a line from the center to any point on the circle. That line is the radius, which is 1. Drop a line straight down to the x-axis, and you get a right triangle with legs x and y and a hypotenuse of 1. So x² + y² = 1². The Pythagorean theorem calculator covers this rule in depth.

Why a radius of 1? Because it makes trigonometry simple. In a right triangle, cos θ = adjacent ÷ hypotenuse and sin θ = opposite ÷ hypotenuse. When the hypotenuse is 1, those ratios are just the lengths of the sides. So the x-coordinate is cos θ, and the y-coordinate is sin θ.

How angles are measured: angles start at the positive x-axis and go counterclockwise. A negative angle goes clockwise. One full trip around the circle is 360°, or 2π radians.

Sine, Cosine And Tangent On The Unit Circle

On the unit circle, cosine is the x-coordinate, sine is the y-coordinate, and tangent is y divided by x.

• cos θ = x
• sin θ = y
• tan θ = y ÷ x = sin θ ÷ cos θ

The other three trig functions are reciprocals of these:

For example, at 60° the point is (1/2, √3/2), so sec 60° = 1 ÷ (1/2) = 2.

When a value is undefined: tangent and secant divide by x, so they are undefined where x = 0, at 90° and 270°. Cotangent and cosecant divide by y, so they are undefined where y = 0, at 0° and 180°.

Unit Circle Chart In Degrees And Radians

This chart lists the 16 standard angles and their exact coordinates. The first coordinate is cos θ, the second is sin θ.

In decimals, √2/2 ≈ 0.7071, √3/2 ≈ 0.8660 and √3/3 ≈ 0.5774. Teachers usually want the exact form.

Notice that only three different numbers appear in the coordinates, apart from 0 and 1: 1/2, √2/2 and √3/2. The rest of the chart is the same three values with different signs.

Where The Special Values Come From

The exact values come from two special right triangles. You can rebuild the whole chart from them if you forget a value.

The 45-45-90 triangle. At 45°, the triangle inside the unit circle has two equal legs, so x = y. Using x² + y² = 1, you get 2x² = 1, so x = √(1/2) = √2/2. That gives the point (√2/2, √2/2).

The 30-60-90 triangle. This triangle is half of an equilateral triangle, so its shortest side is half the hypotenuse. With a hypotenuse of 1, the short side is 1/2. Then the long side is √(1 − 1/4) = √3/2.

• At 30°, the long side runs along the x-axis: (√3/2, 1/2).
• At 60°, the triangle flips, and the long side is vertical: (1/2, √3/2).

A quick check: as the angle grows from 0° to 90°, the x-value should shrink and the y-value should grow. At 30° the point is closer to the x-axis, so its x-value (√3/2) is the larger one.

How To Memorize The Unit Circle

You only need to memorize the first quadrant, from 0° to 90°. The rest of the circle repeats those values with different signs.

The Square Root Pattern

Write the sine values for 0°, 30°, 45°, 60° and 90° as √0/2, √1/2, √2/2, √3/2 and √4/2. Simplified, those are 0, 1/2, √2/2, √3/2 and 1.

Cosine is the same list in reverse: √4/2, √3/2, √2/2, √1/2, √0/2. So you only need to remember one simple pattern, counting 0 to 4 under the square root.

The Hand Trick

Hold up your left hand, palm facing you. Assign your fingers the angles 0° (pinky), 30°, 45°, 60° and 90° (thumb).

1. Fold down the finger for your angle.
2. The number of fingers below the folded finger gives sine: sin θ = √(fingers below) ÷ 2.
3. The number of fingers above the folded finger gives cosine: cos θ = √(fingers above) ÷ 2.

For 30°, fold the ring finger. There is 1 finger below (the pinky) and 3 above, so sin 30° = √1/2 = 1/2 and cos 30° = √3/2.

Quadrant Signs: All Students Take Calculus

The signs of sine, cosine and tangent change by quadrant. The phrase "All Students Take Calculus" tells you which functions are positive, starting in Quadrant I and moving counterclockwise.

How To Draw The Unit Circle From Memory

You can draw the whole unit circle in about two minutes with a simple routine. Practicing this a few times is often faster than memorizing a printed chart.

1. Draw the axes and the circle. Mark the four axis points: (1, 0) at 0°, (0, 1) at 90°, (−1, 0) at 180° and (0, −1) at 270°.
2. Add the first quadrant. Mark 30°, 45° and 60°, and write their points: (√3/2, 1/2), (√2/2, √2/2) and (1/2, √3/2).
3. Reflect into Quadrant II. Each first-quadrant angle has a mirror across the y-axis: 30° pairs with 150°, 45° with 135°, and 60° with 120°. Copy the points and make the x-values negative.
4. Reflect into Quadrants III and IV. Mirror the top half across the x-axis. 150° pairs with 210°, 30° with 330°, and so on. Make the y-values negative.
5. Add radians last. Angles with a reference angle of 30° use sixths of π (π/6, 5π/6, 7π/6, 11π/6), 45° angles use fourths (π/4, 3π/4, 5π/4, 7π/4), and 60° angles use thirds (π/3, 2π/3, 4π/3, 5π/3).

A quick self-check: every point in the same "family," such as all the 30° reference angles, uses the same two numbers. Only the signs change.

Degrees And Radians

Degrees and radians are two ways to measure the same angle. A full circle is 360°, or 2π radians, so π radians = 180°.

To convert degrees to radians, multiply by π/180. For 225°: 225 × π/180 = 5π/4.

To convert radians to degrees, multiply by 180/π. For 2π/3: (2π/3) × (180/π) = 120°.

One radian is the angle where the arc length equals the radius. It is about 57.3°, a little less than 60°. On the unit circle, the radian measure of an angle equals the length of the arc it covers, which is why radians are the natural unit in calculus and physics.

A helpful habit is to think of radians as fractions of a half turn. π/6 is one-sixth of 180°, or 30°, and π/4 is one-quarter, or 45°. If the fraction itself needs simplifying, like 10π/12, the fraction calculator reduces it to 5π/6.

How To Find Any Angle On The Unit Circle

To find the trig values of any angle, find its reference angle, look up the first-quadrant value, and then attach the sign for its quadrant.

The reference angle is the acute angle between the terminal side and the x-axis.

Example: cos 330°

1. 330° is in Quadrant IV.
2. Reference angle: 360° − 330° = 30°.
3. cos 30° = √3/2.
4. Cosine is positive in Quadrant IV, so cos 330° = √3/2.

Example: sin(7π/6)

1. 7π/6 = 210°, which is in Quadrant III.
2. Reference angle: 210° − 180° = 30°.
3. sin 30° = 1/2.
4. Sine is negative in Quadrant III, so sin(7π/6) = −1/2.

Negative angles and angles past 360°. Add or subtract 360° until the angle is between 0° and 360°. This gives a coterminal angle, which lands on the same point. So 405° is the same as 45°, and −60° is the same as 300°.

The Pythagorean Identity

Because every point on the unit circle satisfies x² + y² = 1, and x = cos θ and y = sin θ, you get the most important identity in trigonometry:

sin²θ + cos²θ = 1

This lets you find one trig value from another. Example: sin θ = 3/5, and θ is in Quadrant II.

1. cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25.
2. cos θ = ±4/5.
3. Cosine is negative in Quadrant II, so cos θ = −4/5.

Dividing the identity by cos²θ gives a second form, 1 + tan²θ = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ.

Inverse Trig Functions On The Unit Circle

Inverse trig functions work the unit circle backward. Instead of starting with an angle, you start with a value and find the angle.

Example: arcsin(1/2). Look for the point where the y-coordinate is 1/2. In Quadrant I, that is 30°. So arcsin(1/2) = 30°, or π/6.

Many angles share the same sine, such as 30° and 150°. To give one answer, each inverse function has a set range:

When solving an equation like sin θ = 1/2 for all angles between 0° and 360°, remember both answers: 30° and 150°.

Why The Unit Circle Matters

The unit circle turns trigonometry into geometry you can see, and it explains why sine and cosine are waves.

As a point travels around the circle, its height (sine) rises and falls between −1 and 1. Plotting that height against the angle produces the sine curve. Plotting the x-coordinate produces the cosine curve. Both repeat every 360°, or 2π. That repeating pattern is why sine and cosine describe anything that cycles, from sound and light waves to alternating current and the motion of a Ferris wheel.

The unit circle is also the base for precalculus and calculus topics like trig identities, polar coordinates and derivatives of sine and cosine.

Common Unit Circle Mistakes

Summary

• The unit circle has radius 1, center (0, 0) and equation x² + y² = 1.
• Each point is (cos θ, sin θ), and tan θ = sin θ ÷ cos θ.
• Memorize the first quadrant using the √0/2 to √4/2 pattern, then use signs for the other quadrants.
• Use "All Students Take Calculus" for signs and reference angles for any angle.
• sin²θ + cos²θ = 1 comes directly from the circle's equation.

Next step: sketch the unit circle from memory using the steps above, then check yourself against the chart. For more math help, browse the tools on the Calc Solver homepage.

Frequently Asked Questions