Unit Circle Calculator

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Reviewed and Validated by:

David Chen, PhD Mathematics

Expert in Trigonometry and Numerical Analysis

Instantly determine the coordinates and primary trigonometric ratios (Sine, Cosine, Tangent) for any angle on the Unit Circle.

Unit Circle Calculator

Unit Circle Calculator Formula

The unit circle is defined by the equation $x^2 + y^2 = 1$. The formulas used to find the coordinates and trigonometric ratios for an angle $\theta$ are:

$$ x = \cos(\theta) \\ y = \sin(\theta) \\ \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \\ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

Formula Sources:

Variables

The calculator requires one primary input to solve for the remaining values:

  • Angle ($\theta$) in Degrees: The measure of the angle, counter-clockwise from the positive x-axis, used to determine the point on the circle.

What is Unit Circle Calculator?

The Unit Circle Calculator is a tool designed to quickly find the coordinates (x and y) of a point on the circumference of the unit circle corresponding to a given angle, along with the fundamental trigonometric ratios.

The unit circle is a circle with a radius of one unit centered at the origin (0, 0) of the Cartesian coordinate system. It serves as a visual and foundational tool in trigonometry. For any angle $\theta$, the x-coordinate of the intersection point is $\cos(\theta)$, and the y-coordinate is $\sin(\theta)$. This relationship is the core principle used by this calculator to derive all outputs.

Using a calculator saves time and ensures high precision, especially for angles that are not standard, enabling students, engineers, and mathematicians to quickly verify results or explore trigonometric functions.

How to Calculate Unit Circle Values (Example)

Let’s find the values for an angle of $150^\circ$:

  1. Enter the Angle: Input $150$ into the Angle in Degrees field.
  2. Convert to Radians: The angle in radians is $150 \times \frac{\pi}{180} \approx 2.618$ radians.
  3. Calculate X-Coordinate (Cosine): $x = \cos(150^\circ) = -\frac{\sqrt{3}}{2} \approx -0.8660$.
  4. Calculate Y-Coordinate (Sine): $y = \sin(150^\circ) = \frac{1}{2} = 0.5$.
  5. Calculate Tangent: $\tan(150^\circ) = \frac{0.5}{-0.8660} \approx -0.5774$.

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Frequently Asked Questions (FAQ)

What is the purpose of the unit circle?

The unit circle provides a visual and algebraic framework for defining trigonometric functions for all real numbers, not just acute angles in a right triangle. It shows the periodic nature of sine and cosine.

What is the maximum and minimum value for sine and cosine?

Since the unit circle has a radius of 1, both the x-coordinate (cosine) and the y-coordinate (sine) must fall between -1 and 1, inclusive. The maximum is 1 and the minimum is -1.

How do I convert degrees to radians manually?

To convert an angle from degrees to radians, you multiply the degree measure by the conversion factor $\frac{\pi}{180}$. For example, $90^\circ$ is $90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

Why does the tangent sometimes show “Infinity” or “Undefined”?

Tangent is calculated as $y/x$ or $\sin(\theta)/\cos(\theta)$. It is undefined when the x-coordinate (or cosine value) is zero. This occurs at $90^\circ$ ($\pi/2$ radians) and $270^\circ$ ($3\pi/2$ radians).

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