Unit Circle Calculator

Calculate trigonometric functions (sine, cosine, tangent, etc.) and coordinates for any angle on the unit circle.

Results

Coordinates (cos θ, sin θ): (0.7071, 0.7071)

Results copied!

What is the Unit Circle?

The unit circle is a circle with a radius of exactly 1, centered at the origin (0,0) of the Cartesian coordinate system. Its equation is x² + y² = 1. The unit circle is fundamental in trigonometry because it provides a simple way to define and visualize the trigonometric functions (sine, cosine, tangent, etc.) for all real-numbered angles, including those outside the typical 0° to 90° range of a right triangle.

For any point (x, y) on the circumference of the unit circle, if a radius is drawn from the origin to that point, it forms an angle (θ) with the positive x-axis. The coordinates of that point are directly related to the primary trigonometric functions: the x-coordinate is the cosine of the angle (cos θ), and the y-coordinate is the sine of the angle (sin θ). This relationship holds true for any angle.

Unit Circle Formula and Explanation

The core concept of the unit circle connects angles to coordinates. For any angle θ measured from the positive x-axis, the corresponding point (x, y) on the circle gives the values for cosine and sine.

• cos(θ) = x
• sin(θ) = y
• tan(θ) = y / x

The other three main trigonometric functions are reciprocals of the first three:

• sec(θ) = 1 / x (reciprocal of cosine)
• csc(θ) = 1 / y (reciprocal of sine)
• cot(θ) = x / y (reciprocal of tangent)

Variables Table

Variable	Meaning	Unit	Typical Range
θ (theta)	The angle measured counter-clockwise from the positive x-axis.	Degrees or Radians	Any real number (e.g., 0 to 360° or 0 to 2π rad)
x	The horizontal coordinate of the point on the circle; represents cos(θ).	Unitless	-1 to 1
y	The vertical coordinate of the point on the circle; represents sin(θ).	Unitless	-1 to 1

Practical Examples

Example 1: Calculating for 60 Degrees

Let’s find the trigonometric values for an angle of 60°.

• Input Angle: 60°
• Input Unit: Degrees
• Calculation:
  • cos(60°) = 0.5
  • sin(60°) = √3 / 2 ≈ 0.8660
  • tan(60°) = sin(60°) / cos(60°) = 0.8660 / 0.5 = 1.7320
• Resulting Point (x,y): (0.5, 0.8660)

Example 2: Calculating for 3π/4 Radians

Let’s find the values for an angle of 3π/4 radians (which is equivalent to 135°).

• Input Angle: 2.356 (approx. 3π/4)
• Input Unit: Radians
• Calculation:
  • cos(3π/4) = -√2 / 2 ≈ -0.7071
  • sin(3π/4) = √2 / 2 ≈ 0.7071
  • tan(3π/4) = sin(3π/4) / cos(3π/4) = 0.7071 / -0.7071 = -1
• Resulting Point (x,y): (-0.7071, 0.7071)

How to Use This Unit Circle Calculator

This calculator is designed for ease of use and clarity. Follow these simple steps:

1. Enter the Angle: Type the numerical value of your angle into the “Angle Value” field.
2. Select the Unit: Use the dropdown menu to choose whether your input angle is in “Degrees (°)” or “Radians (rad)”.
3. View the Results: The calculator automatically updates as you type. The results section will immediately display the coordinates (cos θ, sin θ) and a full table of all six trigonometric functions.
4. Interpret the Visualization: The SVG chart provides a visual representation of your angle on the unit circle. The red line shows the angle’s terminal side, and the text displays the (x, y) coordinates at that point.
5. Reset or Copy: Use the “Reset” button to return to the default values or the “Copy Results” button to copy a summary of the calculations to your clipboard.

Key Factors That Affect Unit Circle Values

1. Angle: This is the primary input. The values of all trigonometric functions depend entirely on the angle provided.
2. Quadrant: The quadrant where the angle’s terminal side lies determines the sign (positive or negative) of the sine, cosine, and tangent values. For instance, in Quadrant II (90° to 180°), cosine (x) is negative while sine (y) is positive.
3. Unit of Measurement (Degrees vs. Radians): While the position on the circle is the same, the numerical input value is different. 360 degrees is equivalent to 2π radians. Ensure you select the correct unit for accurate calculations.
4. Reference Angle: This is the acute angle that the terminal side makes with the x-axis. Angles in different quadrants can have the same reference angle, leading to similar values but with different signs.
5. Special Angles (0°, 30°, 45°, 60°, 90°): These angles and their multiples have exact, well-known sine and cosine values involving integers and square roots, which form the building blocks of the unit circle.
6. Periodicity: Trigonometric functions are periodic. This means their values repeat after a certain interval. For sine and cosine, the period is 360° (or 2π radians). Adding or subtracting multiples of this period to an angle will result in the same function values.

Frequently Asked Questions (FAQ)

1. What are the coordinates at 90 degrees?

At 90°, the point on the unit circle is at the very top, which corresponds to the coordinates (0, 1). Therefore, cos(90°) = 0 and sin(90°) = 1.

2. Why is the tangent of 90 degrees undefined?

Tangent is defined as sin(θ) / cos(θ). At 90°, cos(90°) is 0. Since division by zero is undefined, tan(90°) is also undefined.

3. How do I convert from degrees to radians?

To convert from degrees to radians, multiply the degree value by (π / 180). For example, 180° * (π / 180) = π radians. Our angle converter can do this automatically.

4. How do I convert from radians to degrees?

To convert from radians to degrees, multiply the radian value by (180 / π). For example, π/2 rad * (180 / π) = 90°.

5. Can I use a negative angle in the calculator?

Yes. Negative angles are measured clockwise from the positive x-axis. For example, -90° is equivalent to 270°, and the calculator will provide the correct values for this position (cos = 0, sin = -1).

6. What is the relationship between the unit circle and the Pythagorean theorem?

The equation for the unit circle is x² + y² = 1. Since x = cos(θ) and y = sin(θ), substituting these gives (cos θ)² + (sin θ)² = 1. This is the famous Pythagorean Identity, one of the most important identities in trigonometry, derived directly from the Pythagorean theorem applied to the unit circle.

7. What is sine and cosine of 45 degrees?

For 45 degrees, both sine and cosine are equal: sin(45°) = cos(45°) = √2 / 2 ≈ 0.7071. This is a common value to see in a right triangle calculator.

8. What is the unit circle used for?

It’s used to extend trigonometric functions beyond right-angled triangles to any angle, find exact trig values for special angles, solve trig equations, and understand the periodic nature of waves in physics and engineering. Our trigonometry calculator uses these principles.