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The Unit Circle Calculator maps any angle to its corresponding point on the unit circle, providing the coordinates $$(\cos\theta, \sin\theta)$$, the quadrant, the reference angle, and all six trigonometric function values. The unit circle is the central framework for extending trigonometry beyond right triangles to all real numbers.
The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. Every point on the unit circle has coordinates:
$$(x, y) = (\cos\theta, \sin\theta)$$
where $$\theta$$ is the angle measured counterclockwise from the positive x-axis. This definition elegantly extends sine and cosine to all angles, not just those in a right triangle.
The coordinate plane is divided into four quadrants, and the signs of trigonometric functions follow a memorable pattern often summarized as "All Students Take Calculus":
The reference angle is the acute angle between the terminal side of $$\theta$$ and the x-axis. It always lies between 0° and 90°. The trigonometric functions of any angle equal the corresponding functions of its reference angle, with the sign determined by the quadrant.
The most important angles to memorize are 0°, 30°, 45°, 60°, and 90° (and their counterparts in other quadrants). At these angles, the coordinates involve only the values 0, $$\frac{1}{2}$$, $$\frac{\sqrt{2}}{2}$$, $$\frac{\sqrt{3}}{2}$$, and 1 in various combinations.
From the unit circle point $$(x, y)$$, all six trig functions are defined:
$$\sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x}$$
$$\csc\theta = \frac{1}{y}, \quad \sec\theta = \frac{1}{x}, \quad \cot\theta = \frac{x}{y}$$
Enter any angle in degrees (positive, negative, or beyond 360°). The calculator normalizes the angle to the range 0°–360°, identifies the quadrant and reference angle, then computes the unit circle coordinates and all six trigonometric values.
The calculator first normalizes the angle using $$((\theta \bmod 360) + 360) \bmod 360$$ to map it into the range [0°, 360°). It then determines the quadrant from the normalized value and computes the reference angle. Finally, it evaluates all six trigonometric functions using the original angle converted to radians.
The x-coordinate is $$\cos\theta$$ and the y-coordinate is $$\sin\theta$$. Together they represent the point where the terminal side of the angle intersects the unit circle. The quadrant tells you the sign pattern, and the reference angle tells you which standard angle the values relate to. NaN indicates an undefined function at that angle.
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150° is in Quadrant II with a reference angle of 30°. Cosine is negative, sine is positive. The values match the 30° reference: |cos| = √3/2, sin = 1/2.
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−120° normalizes to 240° (Quadrant III, reference angle 60°). Both sine and cosine are negative, but tangent is positive (negative/negative), matching the ASTC rule.
The unit circle is a circle of radius 1 centered at the origin. Each point on it is described by $$(\cos\theta, \sin\theta)$$ for an angle $$\theta$$ measured from the positive x-axis. It provides the standard definitions of all trigonometric functions for any angle.
After normalizing the angle to 0°–360°: Quadrant I is 0°–90°, Quadrant II is 90°–180°, Quadrant III is 180°–270°, and Quadrant IV is 270°–360°. Angles on the axes (0°, 90°, 180°, 270°) are considered quadrantal angles.
A reference angle is the acute angle (0°–90°) formed between the terminal side of the angle and the nearest part of the x-axis. It equals $$\theta$$ in QI, $$180° - \theta$$ in QII, $$\theta - 180°$$ in QIII, and $$360° - \theta$$ in QIV.
ASTC stands for All-Sine-Tangent-Cosine, a mnemonic for which trig functions are positive in each quadrant. In Q-I all are positive, in Q-II only sine, in Q-III only tangent, and in Q-IV only cosine. A common phrase is "All Students Take Calculus."
The unit circle provides a geometric foundation for trigonometry that extends beyond right triangles. It defines trig functions for all angles (including negative angles and angles greater than 360°), connects to complex numbers via Euler's formula $$e^{i\theta} = \cos\theta + i\sin\theta$$, and underlies Fourier analysis.
Angles are periodic with period 360°. To normalize, compute $$((\theta \bmod 360) + 360) \bmod 360$$. For example, 450° normalizes to 90°, and −45° normalizes to 315°. The trigonometric values are the same for all coterminal angles.
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