Unit Circle Chart Explained: Sin, Cos and Tan at Every Angle

The unit circle is a circle of radius 1 centered at the origin, and every point on it is (cos θ, sin θ) for the angle θ that reaches it. Read the x-coordinate and you have the cosine; read the y-coordinate and you have the sine; divide y by x and you have the tangent. So the point at 30° is (√3/2, 1/2), which means cos 30° = √3/2 ≈ 0.866, sin 30° = 1/2 and tan 30° = 1/√3 = √3/3 ≈ 0.577.

You do not need to memorize sixteen separate points. Three numbers, the quadrant signs and a few patterns rebuild the whole chart in under a minute. Below is the full chart with tangent, then how to read any angle from it, then how to remember it for a test. To check any value as you go, open the interactive Unit Circle, which shows exact and decimal values of all six trig functions for any angle you type.

The unit circle chart with sin, cos and tan

These are the 16 standard angles, in degrees and radians, with their exact values. The coordinates of each point are (cos θ, sin θ).

Tangent is undefined at 90° and 270° because x is 0 there, and you cannot divide by zero. The decimals you will see most often are √3/2 ≈ 0.8660, √2/2 ≈ 0.7071 and √3 ≈ 1.7321.

Why the coordinates are cosine and sine

Draw a line from the center to a point on the circle at angle θ, measured counterclockwise from the positive x-axis. Drop a vertical line from that point to the x-axis. You now have a right triangle whose hypotenuse is the radius, 1.

In a right triangle, cosine is adjacent ÷ hypotenuse and sine is opposite ÷ hypotenuse. With a hypotenuse of 1, those become just the adjacent side (the horizontal distance, x) and the opposite side (the vertical distance, y). That is why the unit circle is defined with radius 1: it turns the ratios into plain coordinates. The OpenStax Precalculus textbook defines sine and cosine this way, as the y- and x-coordinates of the point on the unit circle.

Two facts follow straight from the picture:

• Every value of sine and cosine is between −1 and 1, because no point on a circle of radius 1 is farther than 1 from either axis.
• cos²θ + sin²θ = 1 for every angle, because x² + y² = 1 is the equation of the circle. This is the Pythagorean identity, and it lets you find one value from the other.

Tangent is y ÷ x, the slope of the radius line. The other three functions are reciprocals: sec θ = 1/cos θ, csc θ = 1/sin θ and cot θ = 1/tan θ, as covered in the next OpenStax section.

How to read any angle from the unit circle

Every standard angle beyond the first quadrant is a mirror image of one in the first quadrant. Use three steps.

1. Find the quadrant. Quadrant I is 0° to 90°, II is 90° to 180°, III is 180° to 270°, IV is 270° to 360°. If the angle is above 360° or negative, add or subtract 360° (2π) until it is in range. That is its coterminal angle.
2. Find the reference angle, the acute angle between the radius and the x-axis:
   • Quadrant II: 180° − θ
   • Quadrant III: θ − 180°
   • Quadrant IV: 360° − θ
3. Take the first-quadrant values for the reference angle, then fix the signs. In quadrant II, x is negative; in III, both are negative; in IV, y is negative.

Three worked examples:

• sin 210°. Quadrant III, reference angle 210° − 180° = 30°. sin 30° = 1/2, and y is negative in quadrant III, so sin 210° = −1/2.
• cos 5π/3. 5π/3 is 300°, in quadrant IV, with reference angle 60°. cos 60° = 1/2, and x is positive in quadrant IV, so cos 5π/3 = 1/2.
• tan 135°. Quadrant II, reference angle 45°. tan 45° = 1, and tangent is negative in quadrant II (x negative, y positive), so tan 135° = −1.

If you switch between degrees and radians often, the Degrees to Radians Converter does it exactly: multiply degrees by π/180 to get radians.

How to memorize the unit circle

You only need the first quadrant. Everything else is the same three values with signs.

Pattern 1: the square-root count

Write sine for 0°, 30°, 45°, 60° and 90° as √0/2, √1/2, √2/2, √3/2, √4/2. Simplified, that is 0, 1/2, √2/2, √3/2, 1. Cosine is the same list backward: 1, √3/2, √2/2, 1/2, 0.

Pattern 2: All Students Take Calculus

Going counterclockwise from quadrant I, the functions that are positive are All, Sine, Tangent, Cosine:

Pattern 3: radian denominators

Each standard radian angle tells you its reference angle by its denominator:

• Denominator 6 (π/6, 5π/6, 7π/6, 11π/6): reference angle 30°, close to the x-axis. The point is wide and low, so |cos| = √3/2 and |sin| = 1/2.
• Denominator 4 (π/4, 3π/4, 5π/4, 7π/4): reference angle 45°, on a diagonal. |cos| = |sin| = √2/2.
• Denominator 3 (π/3, 2π/3, 4π/3, 5π/3): reference angle 60°, close to the y-axis. The point is narrow and tall, so |cos| = 1/2 and |sin| = √3/2.

Then the numerator places it in a quadrant. For example, 7π/6 is a little past π (6π/6), so it is in quadrant III, and both coordinates are negative: (−√3/2, −1/2).

Pattern 4: the two triangles

All of these values come from two triangles. The 45-45-90 triangle has sides in the ratio 1 : 1 : √2, so scaling the hypotenuse to 1 gives legs of 1/√2 = √2/2. The 30-60-90 triangle has sides 1 : √3 : 2, so scaling the hypotenuse to 1 gives legs of 1/2 and √3/2. If you forget a value, sketch the triangle. The Triangle Solver confirms the sides and angles of any triangle you sketch.

Common mistakes

• Swapping sine and cosine. The coordinates are (cos, sin), in alphabetical order, like (x, y). A point near the x-axis has a large x, so a small angle has a large cosine.
• Forgetting the sign. Take the value from the reference angle, then check the quadrant every time.
• Calculator in the wrong mode. sin 30 is 0.5 in degree mode and about −0.988 in radian mode, because the calculator reads 30 as 30 radians.
• Mixing up the arc and the angle. On the unit circle an angle of θ radians cuts off an arc of length θ, because arc length = radius × angle and the radius is 1. The Circle Calculator shows the arc length for any radius and angle.

Common questions

What is the unit circle used for? It defines sine, cosine and tangent for every angle, including angles above 90° and negative angles, where right-triangle definitions stop working. It is the basis for graphing trig functions, solving trig equations and working with waves and rotations.

Why is the radius of the unit circle 1? With a radius of 1, the hypotenuse of every triangle in the circle is 1, so cosine and sine equal the x- and y-coordinates directly, with no division.

What is tan on the unit circle? Tangent is the y-coordinate divided by the x-coordinate, which is the slope of the radius. It is undefined at 90° and 270°, where x is 0.

How long does it take to memorize the unit circle? With the square-root pattern and All Students Take Calculus, most students can rebuild the chart from memory after a few practice sessions, since only three values (1/2, √2/2, √3/2) ever appear apart from 0 and 1.

What is the unit circle value of sin 45°? sin 45° = cos 45° = √2/2 ≈ 0.7071, because the point at 45° sits on the diagonal where x equals y.