Unit Circle Reference

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Click any angle on the circle to see its exact trigonometric values, reference triangle, and coordinates.

°
Deg Rad cos θ sin θ tan θ
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About This Tool

The unit circle is a circle of radius 1 centered at the origin. It is the fundamental visual tool in trigonometry, connecting angles to their sine, cosine, and tangent values through the coordinates of points on the circle. This interactive reference displays all 16 special angles with their exact values, reference triangles, and quadrant sign rules. Enter any custom angle to see its approximate values.

Sources: Wikipedia · Math is Fun

How to Use

  1. Step 1 Click any angle dot on the circle, or type a custom angle in the input field.
  2. Step 2 View the exact trigonometric values, reference triangle, and quadrant sign in the panel below.
  3. Step 3 Toggle between degrees and radians, or expand the reference table and ASTC chart for quick lookup.

Methodology

The unit circle is defined as x² + y² = 1. For any angle θ measured counterclockwise from the positive x-axis, the terminal point on the circle has coordinates (cos θ, sin θ). All other trig functions derive from these: tan θ = sin θ / cos θ, and the reciprocal functions are csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. The 16 special angles come from 30-60-90 and 45-45-90 reference triangles placed in each quadrant. The ASTC rule determines the sign of each function based on the quadrant.

Sources: Wikipedia — Unit circle · Khan Academy

Understanding Your Results

When you select an angle, the tool shows its position on the circle with a radius line from the origin. The x-coordinate of the point is the cosine, and the y-coordinate is the sine. The dashed reference triangle shows how the angle relates to the x-axis. Special angles display exact values (like √2/2), while other angles show decimal approximations. The quadrant indicator tells you which trig functions are positive at that angle.

Sources

Practical Examples

Example 1: Finding sin(135°) The reference angle for 135° is 180° − 135° = 45°. Since 135° is in Quadrant II, sine is positive. Therefore, sin(135°) = sin(45°) = √2/2 ≈ 0.7071. Example 2: Finding cos(240°) The reference angle for 240° is 240° − 180° = 60°. Since 240° is in Quadrant III, cosine is negative. Therefore, cos(240°) = −cos(60°) = −1/2 = −0.5.

Memorization Tips

1. Learn only the first quadrant (0° to 90°) — all other quadrants mirror these values with sign changes determined by the ASTC rule. 2. The sine pattern for 0°, 30°, 45°, 60°, 90° is √0/2, √1/2, √2/2, √3/2, √4/2, which simplifies to 0, 1/2, √2/2, √3/2, 1. Cosine is the same sequence reversed. 3. Use the "hand trick": hold your left hand palm up, assign 0°, 30°, 45°, 60°, 90° to each finger, fold down the finger for your angle, and count remaining fingers to get sin and cos. 4. Remember tangent as sine divided by cosine — you can always derive it from the other two values. 5. The reference triangle always connects the point on the circle to the x-axis (never the y-axis), forming a right triangle.

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Frequently Asked Questions
What is the unit circle?
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of the coordinate plane. Every point on the circle corresponds to an angle, and its coordinates give the cosine (x) and sine (y) of that angle. It is a fundamental tool in trigonometry for understanding how trigonometric functions relate to angles.
What are the 16 special angles on the unit circle?
The 16 special angles are multiples of 30° and 45° from 0° to 330°: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°. These angles have exact trigonometric values involving simple fractions and square roots (√2 and √3), making them the foundation of most trigonometry problems.
What is the difference between degrees and radians?
Degrees and radians are two ways to measure angles. A full circle is 360° or 2π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 90° = π/2 radians, and 180° = π radians. Radians are preferred in advanced mathematics because they simplify many formulas.
What is a reference angle?
A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of an angle and the x-axis. It helps determine exact trig values for angles in any quadrant by relating them back to first-quadrant values. For Quadrant II, the reference angle is 180° − θ. For Quadrant III, it is θ − 180°. For Quadrant IV, it is 360° − θ.
What does the ASTC rule (All Students Take Calculus) mean?
ASTC is a mnemonic for remembering which trigonometric functions are positive in each quadrant. A = All (Quadrant I: all functions positive), S = Sine (Quadrant II: only sine and cosecant positive), T = Tangent (Quadrant III: only tangent and cotangent positive), C = Cosine (Quadrant IV: only cosine and secant positive). The phrase "All Students Take Calculus" makes it easy to remember.
How do the 30-60-90 and 45-45-90 triangles relate to the unit circle?
The special right triangles are the source of the exact values on the unit circle. The 45-45-90 triangle has sides in ratio 1 : 1 : √2, giving coordinates (√2/2, √2/2) at 45°. The 30-60-90 triangle has sides in ratio 1 : √3 : 2, giving coordinates (√3/2, 1/2) at 30° and (1/2, √3/2) at 60°. By "dropping" these reference triangles onto the unit circle in each quadrant, you get all 16 special angle values.
How can I memorize the unit circle values?
Focus on the first quadrant only (0° to 90°), since all other quadrants mirror these values with sign changes. The sine values for 0°, 30°, 45°, 60°, 90° follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2 — which simplifies to 0, 1/2, √2/2, √3/2, 1. Cosine values are the same sequence in reverse. Use the ASTC mnemonic to determine signs in other quadrants.
Why are some trigonometric values undefined?
A trigonometric function is undefined when its calculation involves division by zero. Tangent (sin/cos) is undefined at 90° and 270° because cosine equals zero there. Similarly, cotangent (cos/sin) is undefined at 0° and 180° where sine equals zero. Cosecant (1/sin) is undefined where sine is zero, and secant (1/cos) is undefined where cosine is zero.
What are practical applications of the unit circle?
The unit circle is used throughout science and engineering. Physics uses it for analyzing circular motion, waves, and oscillations. Engineering applies it in signal processing, AC circuit analysis, and structural design. Computer graphics rely on it for rotations, animations, and 3D transformations. Navigation and astronomy use it for calculating positions and trajectories. Music theory uses it to model sound wave frequencies and harmonics.
How do coordinates on the unit circle relate to sine and cosine?
For any angle θ on the unit circle, the point where the terminal side intersects the circle has coordinates (cos θ, sin θ). The x-coordinate is always the cosine, and the y-coordinate is always the sine. This is the defining relationship of the unit circle and explains why cosine and sine values range from −1 to 1, since points on the unit circle never exceed a distance of 1 from the origin.