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Calculator · math

Unit Circle Calculator

Compute sin, cos, tan, csc, sec, and cot for any angle on the unit circle. Supports degrees and radians with instant, accurate results.

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Inputs

Angle

Angle Unit

Function / Coordinate

Trig Function Value

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Trig Function Value—

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What Is the Unit Circle?

The unit circle is a circle with radius 1 centered at the origin of a Cartesian coordinate plane. Every point P on this circle satisfies the equation x² + y² = 1. For any angle θ measured counterclockwise from the positive x-axis, the coordinates of point P are defined as (cosθ, sinθ). This elegant definition extends trigonometry beyond acute angles in right triangles to all real-number inputs, enabling the analysis of periodic phenomena across physics, engineering, and computer science.

According to Khan Academy’s unit circle series, this approach generalizes sine and cosine beyond the 0°–90° range, giving them meaning for angles in all four quadrants, for negative angles, and for angles greater than 360°.

The Six Trigonometric Functions

Given a point P(θ) = (x, y) on the unit circle, all six trigonometric functions are defined as follows:

sinθ = y — the y-coordinate of P
cosθ = x — the x-coordinate of P
tanθ = sinθ / cosθ = y / x — undefined when x = 0, i.e., at 90° and 270°
cscθ = 1 / sinθ = 1 / y — undefined when y = 0, i.e., at 0° and 180°
secθ = 1 / cosθ = 1 / x — undefined when x = 0, i.e., at 90° and 270°
cotθ = cosθ / sinθ = x / y — undefined when y = 0, i.e., at 0° and 180°

Key Angle Values on the Unit Circle

The following standard angles appear repeatedly in mathematics and applied sciences. Exact values are computed directly from the unit circle geometry using 30-60-90 and 45-45-90 triangle ratios:

0° (0 rad): cos = 1, sin = 0, tan = 0
30° (π/6 rad): cos = √3/2 ≈ 0.8660, sin = 1/2 = 0.5000, tan = 1/√3 ≈ 0.5774
45° (π/4 rad): cos = √2/2 ≈ 0.7071, sin = √2/2 ≈ 0.7071, tan = 1.0000
60° (π/3 rad): cos = 1/2 = 0.5000, sin = √3/2 ≈ 0.8660, tan = √3 ≈ 1.7321
90° (π/2 rad): cos = 0, sin = 1, tan = undefined
180° (π rad): cos = −1, sin = 0, tan = 0
270° (3π/2 rad): cos = 0, sin = −1, tan = undefined
360° (2π rad): cos = 1, sin = 0, tan = 0

Degrees vs. Radians

Angles are expressed in either degrees or radians. The exact conversion formula is: radians = degrees × (π / 180). For example, 180° equals π radians (≈ 3.14159), and 90° equals π/2 radians (≈ 1.5708). Radians are the preferred unit in calculus because derivative formulas — such as d/dθ(sinθ) = cosθ — are valid only when θ is measured in radians. As established in The Trigonometric Functions (GVSU ScholarWorks), radian measure arises naturally from arc length: on a unit circle, an arc of length s subtends an angle of exactly s radians at the center.

How This Calculator Works

Enter any angle value, select degrees or radians, and choose which function or coordinate to compute. The calculator evaluates P(θ) = (cosθ, sinθ) and derives all six trigonometric functions from that point. For θ = 60°, the output is: cos = 0.5000, sin = 0.8660, tan = 1.7321, csc = 1.1547, sec = 2.0000, cot = 0.5774. For θ = 3π/4 rad (135°), the output is: cos = −0.7071, sin = 0.7071, tan = −1.0000, csc = 1.4142, sec = −1.4142, cot = −1.0000.

The Four Quadrants and Sign Patterns

The sign of each trigonometric function depends on which quadrant the terminal ray falls in. In Quadrant I (0°–90°), all functions are positive. In Quadrant II (90°–180°), only sine and cosecant are positive. In Quadrant III (180°–270°), only tangent and cotangent are positive. In Quadrant IV (270°–360°), only cosine and secant are positive. The mnemonic ASTC (All Students Take Calculus) encodes this pattern by quadrant.

Real-World Applications

The unit circle provides the mathematical foundation for numerous real-world systems. In electrical engineering, AC voltage is modeled as V(t) = V₀ × sin(2πft + φ), where φ is a phase angle on the unit circle. In physics, projectile motion decomposes initial velocity v₀ into horizontal component v₀cosθ and vertical component v₀sinθ. In computer graphics, 2D rotation matrices use cosθ and sinθ to rotate vectors about the origin. In signal processing, Fourier analysis decomposes complex waveforms into sine and cosine components derived directly from the unit circle.

Methodology & Sources

This calculator’s computations follow the standard definition of trigonometric functions via the unit circle. Primary references include the Khan Academy Unit Circle series and The Trigonometric Functions from GVSU ScholarWorks, both of which provide rigorous derivations of the unit circle definition and its relationship to all six trigonometric functions.

Reference

Frequently asked questions

What is a unit circle calculator and what does it compute?

A unit circle calculator computes the coordinates and all six trigonometric function values for any given angle on the unit circle. Given an angle in degrees or radians, it returns the x-coordinate (cosine), y-coordinate (sine), tangent, cosecant, secant, and cotangent. For example, entering 45 degrees yields sin = 0.7071, cos = 0.7071, tan = 1.0000, csc = 1.4142, sec = 1.4142, and cot = 1.0000, all computed from the unit circle point (root 2 over 2, root 2 over 2).

How do you find exact sin and cos values on the unit circle at 30, 45, and 60 degrees?

At 30 degrees (pi/6 radians), sin = 1/2 (exactly 0.5) and cos = root 3 divided by 2, approximately 0.8660. At 45 degrees (pi/4 radians), both sin and cos equal root 2 divided by 2, approximately 0.7071. At 60 degrees (pi/3 radians), sin = root 3 divided by 2 approximately 0.8660 and cos = 1/2 exactly. These values derive from 30-60-90 and 45-45-90 right triangle ratios projected onto the radius-1 circle.

What is the difference between degrees and radians on the unit circle?

Degrees divide a full rotation into 360 equal parts, while radians measure the arc length swept on the unit circle. Since the unit circle has circumference 2*pi, a full 360-degree rotation equals exactly 2*pi radians (approximately 6.2832). Convert between them using: radians = degrees times (pi / 180). Therefore 90 degrees equals pi/2 radians (approximately 1.5708), 45 degrees equals pi/4 radians, and 180 degrees equals pi radians. Radians are mandatory for calculus derivative and integral formulas involving trig functions.

Why does the tangent function become undefined at 90 and 270 degrees?

Tangent is defined as sin(theta) divided by cos(theta). At 90 degrees (pi/2 radians), the unit circle point is (0, 1), making cos(theta) = 0. Division by zero is mathematically undefined, so tan(90 degrees) does not exist. The same situation occurs at 270 degrees (3*pi/2 radians), where the point is (0, -1) and cos is again zero. On a graph of tangent, these values correspond to vertical asymptotes appearing at every odd multiple of 90 degrees (pi/2 radians).

How is the unit circle used in real-world applications?

The unit circle underlies many real-world systems. Electrical engineers model AC voltage as V(t) = V0 * sin(2*pi*f*t + phi), where phi is a phase angle on the unit circle. Physicists decompose projectile velocity into horizontal (v0 * cos theta) and vertical (v0 * sin theta) components. Computer graphics engines rotate 2D and 3D objects using rotation matrices built from cosine and sine values. Audio and signal engineers use Fourier transforms, which represent complex waveforms as sums of sine and cosine waves, all derived from unit circle geometry.

What are the four quadrants of the unit circle and which trig functions are positive in each?

The unit circle is divided into four quadrants by the x and y axes. In Quadrant I (0 to 90 degrees), all six trig functions are positive. In Quadrant II (90 to 180 degrees), only sine and its reciprocal cosecant are positive. In Quadrant III (180 to 270 degrees), only tangent and its reciprocal cotangent are positive. In Quadrant IV (270 to 360 degrees), only cosine and its reciprocal secant are positive. The mnemonic ASTC, standing for All Students Take Calculus, encodes which functions are positive going counterclockwise through each quadrant.