Unit Circle Calculator - Sin, Cos, Tan Values & Trig Reference Chart
Interactive unit circle calculator. Find exact Sin, Cos, and Tan values for any angle in degrees or radians. Master trigonometry with this free visual tool.
Quick Angles
Quick Coordinates (x, y)
Interactive Unit Circle
Click and drag the point on the circle to change the angleSpecial Angles Reference
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | undefined | 1 | undefined |
| 30° | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | 1 | 0 | undefined | 1 | undefined | 0 |
| 120° | √3/2 | -1/2 | -√3 | 2√3/3 | -2 | -√3/3 |
| 135° | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150° | 1/2 | -√3/2 | -√3/3 | 2 | -2√3/3 | -√3 |
| 180° | 0 | -1 | 0 | undefined | -1 | undefined |
| 210° | -1/2 | -√3/2 | √3/3 | -2 | -2√3/3 | √3 |
| 225° | -√2/2 | -√2/2 | 1 | -√2 | -√2 | 1 |
| 240° | -√3/2 | -1/2 | √3 | -2√3/3 | -2 | √3/3 |
| 270° | -1 | 0 | undefined | -1 | undefined | 0 |
| 300° | -√3/2 | 1/2 | -√3 | -2√3/3 | 2 | -√3/3 |
| 315° | -√2/2 | √2/2 | -1 | -√2 | √2 | -1 |
| 330° | -1/2 | √3/2 | -√3/3 | -2 | 2√3/3 | -√3 |
| 360° | 0 | 1 | 0 | undefined | 1 | undefined |
This free trigonometry tool helps students, teachers, and professionals find exact values for all six trigonometric functions using the unit circle. The interactive visualization displays sine, cosine, tangent, cosecant, secant, and cotangent values for any angle entered in degrees or radians. The visual representation makes it easy to understand trigonometric relationships, identify quadrants, and find reference angles. Perfect for homework, test preparation, or quick reference, this tool provides both decimal approximations and exact values for special angles. Whether learning trigonometry for the first time or needing a reliable reference, this resource serves as your complete unit circle guide.
How to Use Unit Circle Calculator
How to Enter Angles
Start by selecting your preferred angle unit from the dropdown menu. Choose degrees, radians, or gradians depending on your needs, then enter your angle value in the input field. The tool accepts any angle, including negative angles and angles greater than 360 degrees, which are automatically normalized. You can also use the quick angle buttons to instantly select common angles like 30°, 45°, 60°, or 90°. Results update automatically as you type, providing real-time trigonometric function values.
Understanding the Interactive Unit Circle Visualization
The interactive visualization shows a graphical representation of your angle on the unit circle. Click and drag the point on the circle's circumference to change the angle interactively. The visualization displays coordinate axes, special angle markers, and quadrant labels to help you understand the angle's position. The angle line extends from the center to the point on the circle, making it easy to visualize the relationship between the angle and its trigonometric values. This visual approach helps students understand how angles correspond to coordinates and trigonometric function values.
Reading Trigonometric Function Values
After entering an angle, the tool displays all six trigonometric functions: sin, cos, tan, csc, sec, and cot. Each value is shown with high precision, and for special angles, exact values are displayed in radical form. For example, sin(45°) shows both the decimal approximation (0.707107) and the exact value (√2/2). The results table includes copy buttons for each value, making it easy to use the results in other calculations or documents. The tool also shows which quadrant your angle is in and provides the reference angle, which is always between 0 and 90 degrees.
Using Exact Values and Reference Angles
This tool automatically identifies when your angle matches a special angle with exact values. These include angles like 0°, 30°, 45°, 60°, 90°, and their multiples. When an exact value is available, it's displayed alongside the decimal approximation. The reference angle feature helps you understand how any angle relates to its corresponding acute angle in the first quadrant. This is particularly useful for understanding the ASTC mnemonic (All Students Take Calculus), which helps remember which trigonometric functions are positive in each quadrant.
Calculator Features
Interactive Unit Circle Visualization
Click and drag the point on the circle to change angles interactively with real-time updates
All Six Trigonometric Functions
Calculate sin, cos, tan, csc, sec, and cot values for any angle with high precision
Exact Values for Special Angles
Automatically displays exact values in radical form for special angles like 30°, 45°, 60°, and 90°
Reference Angle Calculator
Automatically calculates the reference angle and identifies which quadrant your angle is in
Quadrant Identification
Shows which quadrant your angle belongs to with the ASTC mnemonic for sign rules
Degree and Radian Conversion
Convert between degrees, radians, and gradians with automatic unit conversion
Quick Angle Selection
One-click buttons for common angles like 0°, 30°, 45°, 60°, 90°, 180°, and 270°
Copy to Clipboard
One-click copy for any calculated value with visual confirmation
Special Angles Reference Table
Complete reference table showing exact values for all special angles from 0° to 360°
Mobile-Friendly Design
Fully responsive interface that works perfectly on phones, tablets, and desktop computers
Complete Function List
- Interactive SVG unit circle visualization:
- Click and drag to change angles:
- Real-time angle input in degrees, radians, or gradians:
- All six trigonometric functions calculated:
- Sine (sin) function values:
- Cosine (cos) function values:
- Tangent (tan) function values:
- Cosecant (csc) function values:
- Secant (sec) function values:
- Cotangent (cot) function values:
- Exact values displayed for special angles:
- Reference angle calculation:
- Quadrant identification (I, II, III, IV, or axis):
- ASTC mnemonic display:
- Coordinate display (x, y) on unit circle:
- Arc length calculation:
- Sector area calculation:
- Chord length calculation:
- Angle normalization for values > 360° or negative:
- Quick angle selection buttons:
- Special angles reference table:
- Copy to clipboard for all values:
- Mobile-optimized touch interactions:
- Responsive grid layout:
- Auto-calculation on input change:
Common Calculations & Examples
Example 1: Finding sin, cos, tan for 30 degrees
Problem: Find the values of sin(30°), cos(30°), and tan(30°) using the unit circle
Steps:
- Select "Degrees" as the angle unit
- Enter 30 in the angle input field, or click the "30°" quick angle button
- View the results in the trigonometric functions table
Explanation: The tool shows that 30 degrees is in the first quadrant. The reference angle is 30° itself since it's already in the first quadrant. Results display both decimal approximations and exact values. For 30°, sin equals 1/2, cos equals √3/2, and tan equals √3/3. These are fundamental values that appear frequently in trigonometry problems. The interactive visualization shows the point on the unit circle at coordinates (√3/2, 1/2), which corresponds to these trigonometric values.
Example 2: Converting between degrees and radians
Problem: Convert 45 degrees to radians and find all trigonometric function values
Steps:
- Enter 45 in the angle field with "Degrees" selected
- Note the angle display shows: 45° = 0.7854 rad = 50 grad
- View all six trigonometric function values
Explanation: This tool automatically converts between angle units. When you enter 45 degrees, it shows the equivalent in radians (π/4 ≈ 0.7854) and gradians (50 grad). For 45 degrees, all trigonometric functions have exact values. Both sin and cos equal √2/2, tan equals 1, and the reciprocal functions (csc, sec, cot) are the inverses. This angle is particularly important because it's one of the special angles that appears frequently in trigonometry problems. The visualization shows the point at coordinates (√2/2, √2/2) on the unit circle.
Example 3: Finding reference angles
Problem: Find the reference angle and trigonometric values for 150 degrees
Steps:
- Enter 150 in the angle input field
- View the reference angle in the results section
- Check the quadrant identification
Explanation: This tool automatically finds the reference angle for any angle. For 150 degrees, the reference angle is 30° because 150° is in the second quadrant, and 180° - 150° = 30°. Results show that 150° is in Quadrant II, where only sine is positive (ASTC: All Students Take Calculus). The trigonometric values for 150° are related to the values for 30°, but with appropriate sign changes. Sin(150°) equals sin(30°) = 0.5, cos(150°) equals -cos(30°) = -√3/2, and tan(150°) equals -tan(30°) = -√3/3.
Example 4: Identifying quadrants
Problem: Determine which quadrant 225 degrees is in and find its trigonometric values
Steps:
- Enter 225 in the angle field
- Check the quadrant information in the results
- Review the ASTC mnemonic explanation
Explanation: This tool identifies that 225 degrees is in the third quadrant. The reference angle is 45° because 225° - 180° = 45°. In Quadrant III, only tangent is positive according to the ASTC mnemonic. The trigonometric values for 225° are related to 45°, but both sin and cos are negative. Sin(225°) = -sin(45°) = -√2/2, cos(225°) = -cos(45°) = -√2/2, and tan(225°) = tan(45°) = 1. The visualization shows the point in the lower-left quadrant of the unit circle.
Example 5: Using exact values
Problem: Find the exact values for all trigonometric functions of 60 degrees
Steps:
- Click the "60°" quick angle button or enter 60 manually
- View the results table which shows both decimal and exact values
- Note the exact value column shows radical forms
Explanation: This tool recognizes 60 degrees as a special angle and displays exact values in radical form. These exact values are more precise than decimal approximations and are often required in mathematical work. For 60 degrees, sin equals √3/2, cos equals 1/2, and tan equals √3. The reciprocal functions are calculated as the inverses: csc = 2√3/3, sec = 2, and cot = √3/3. These exact values are fundamental in trigonometry and appear frequently in problems involving 30-60-90 triangles.
Example 6: Working with negative angles
Problem: Calculate trigonometric values for -45 degrees
Steps:
- Enter -45 in the angle input field
- The tool automatically normalizes the angle
- View the results for the equivalent positive angle
Explanation: This tool handles negative angles by normalizing them to the 0-360 degree range. A negative angle of -45° is equivalent to 360° - 45° = 315°. Results show this normalization and calculate all trigonometric values accordingly. For -45° (or 315°), the point is in the fourth quadrant, where only cosine is positive. The values are sin = -√2/2, cos = √2/2, and tan = -1. This demonstrates how the tool simplifies working with angles in any form.