Unit Circle Calculator — Exact Trig Values
Find exact and decimal values of sin, cos, tan, csc, sec, and cot for any angle on the unit circle. Supports degrees and radians, shows reference angles, quadrant, and coordinates.
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What Is the Unit Circle Calculator — Exact Trig Values?
This calculator evaluates all six trigonometric functions for any angle on the unit circle, showing both exact values (for common special angles) and decimal approximations. An interactive SVG diagram plots the angle in real time so you can see exactly where the point sits on the circle.
- ›Degrees and radians — switch freely between units; the calculator converts internally so you always get the right answer.
- ›Exact values for special angles — for multiples of 30°, 45°, and 60° the calculator shows the simplified radical form (e.g. √3/2) alongside the decimal, not just a rounded number.
- ›All six trig functions — sin, cos, tan, csc, sec, and cot are computed simultaneously, with undefined values clearly flagged.
- ›Quadrant identification — the result panel reports the quadrant (I–IV) and reference angle so you can verify sign rules immediately.
- ›SVG visual diagram — shows the angle, the radius line, and dashed coordinate projections on a labelled unit circle so the geometry is always visible.
Formula
Unit Circle Definition
x² + y² = 1
Point P on unit circle = (cos θ, sin θ)
The Six Trigonometric Functions
sin θ = y cos θ = x tan θ = y / x (x ≠ 0)
csc θ = 1 / y (y ≠ 0) sec θ = 1 / x (x ≠ 0) cot θ = x / y (y ≠ 0)
Pythagorean Identity
sin²θ + cos²θ = 1
Reference Angle by Quadrant
Q I: θ Q II: 180° − θ Q III: θ − 180° Q IV: 360° − θ
| θ (deg) | θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undef. |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −1/√3 |
| 180° | π | 0 | −1 | 0 |
| 210° | 7π/6 | −1/2 | −√3/2 | 1/√3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 270° | 3π/2 | −1 | 0 | Undef. |
| 300° | 5π/3 | −√3/2 | 1/2 | −√3 |
| 315° | 7π/4 | −√2/2 | √2/2 | −1 |
| 330° | 11π/6 | −1/2 | √3/2 | −1/√3 |
| 360° | 2π | 0 | 1 | 0 |
| Symbol | Name | Description |
|---|---|---|
| θ | Angle | Input angle — can be entered in degrees or radians; any real number accepted |
| (x, y) | Coordinates | The point on the unit circle: x = cos θ, y = sin θ |
| sin θ | Sine | y-coordinate of the point P on the unit circle |
| cos θ | Cosine | x-coordinate of the point P on the unit circle |
| tan θ | Tangent | Ratio y/x; undefined when cos θ = 0 (θ = 90°, 270°, …) |
| csc θ | Cosecant | Reciprocal of sine (1/sin θ); undefined when sin θ = 0 |
| sec θ | Secant | Reciprocal of cosine (1/cos θ); undefined when cos θ = 0 |
| cot θ | Cotangent | Ratio x/y; undefined when sin θ = 0 (θ = 0°, 180°, …) |
How to Use
- 1Enter the angle: Type any angle — whole numbers, decimals, or negatives are all accepted.
- 2Select the unit: Choose Degrees or Radians using the toggle next to the input field.
- 3Use a quick-angle button: Click any of the preset angle buttons (0°, 30°, 45°, … 360°) to jump straight to a common value.
- 4Press Enter or click Calculate: The results grid, quadrant info, and SVG diagram all update instantly.
- 5Read the results: The six trig values appear with their exact form (if applicable) and decimal approximation. Undefined values are labelled clearly. The SVG shows the point (cos θ, sin θ) on the circle.
Example Calculation
Example: θ = 150°
Given: θ = 150°
Step 1: Identify the quadrant
150° is between 90° and 180° → Quadrant II
Step 2: Find the reference angle
ref = 180° − 150° = 30°
Step 3: Evaluate trig functions (Q II: sin+, cos−, tan−)
sin 150° = sin 30° = 1/2 ≈ 0.5000
cos 150° = −cos 30° = −√3/2 ≈ −0.8660
tan 150° = −tan 30° = −1/√3 ≈ −0.5774
Step 4: Reciprocal functions
csc 150° = 1/sin 150° = 2
sec 150° = 1/cos 150° = −2/√3 ≈ −1.1547
cot 150° = 1/tan 150° = −√3 ≈ −1.7321
Point P = (−√3/2, 1/2) ≈ (−0.8660, 0.5000)
Sign rule in Quadrant II (S — sine positive)
In Quadrant II, only sine (and its reciprocal csc) are positive. Cosine, tangent, secant, and cotangent are all negative. The ASTC mnemonic — All, Sine, Tangent, Cosine — tells you which functions are positive in each quadrant.
Understanding Unit Circle — Exact Trig Values
What Is the Unit Circle?
The unit circle is a circle of radius 1 centred at the origin of the Cartesian plane. Its equation is x² + y² = 1. Every point P on the circle can be written as (cos θ, sin θ) where θ is the angle measured counterclockwise from the positive x-axis. This elegant relationship means the unit circle is the foundation of all trigonometry — it defines sine and cosine geometrically, without needing a right triangle.
Because the radius is always 1, the coordinates of the point directly give you the sine and cosine values: x = cos θ and y = sin θ. The four remaining trig functions follow as ratios and reciprocals of these two.
The Four Quadrants and Sign Rules
The coordinate axes divide the unit circle into four quadrants, and the signs of the trig functions change from quadrant to quadrant. The mnemonic ASTC ("All Students Take Calculus") tells you which functions are positive in each:
| Quadrant | Angle range | Positive functions | Negative functions |
|---|---|---|---|
| I | 0° – 90° | All (sin, cos, tan, csc, sec, cot) | — |
| II | 90° – 180° | sin, csc | cos, tan, sec, cot |
| III | 180° – 270° | tan, cot | sin, cos, csc, sec |
| IV | 270° – 360° | cos, sec | sin, tan, csc, cot |
Special Angles and Exact Values
The angles 0°, 30°, 45°, 60°, and 90° (and their quadrant equivalents) have exact trig values that can be written with simple fractions and square roots. These arise from the geometry of equilateral triangles (30–60–90) and isoceles right triangles (45–45–90). Memorising or recognising these values is essential for calculus and physics.
- ›0° / 360°: sin = 0, cos = 1, tan = 0. Point is at (1, 0).
- ›30° (π/6): sin = 1/2, cos = √3/2, tan = 1/√3 = √3/3.
- ›45° (π/4): sin = cos = √2/2 ≈ 0.7071. The point is on the 45° diagonal.
- ›60° (π/3): sin = √3/2, cos = 1/2, tan = √3.
- ›90° (π/2): sin = 1, cos = 0, tan undefined. Point is at (0, 1).
Converting Between Degrees and Radians
Degrees and radians are two ways to measure the same angle. A full circle is 360° = 2π radians, so the conversion factors are:
Degrees → Radians: rad = deg × π / 180
Radians → Degrees: deg = rad × 180 / π
Example: 150° × π/180 = 5π/6 ≈ 2.6180 rad
Example: π/3 × 180/π = 60°
In higher mathematics, radians are the natural unit because they make derivatives and integrals of trig functions simpler (d/dx sin x = cos x only holds when x is in radians). The unit circle calculator accepts both, converting internally as needed.
Real-World Applications of the Unit Circle
- ›Signal processing: Sine and cosine waves are the fundamental components of any periodic signal. Fourier analysis decomposes any waveform into unit-circle coordinates.
- ›Rotational mechanics: Angular velocity, centripetal acceleration, and torque all use trigonometric functions derived from unit-circle relationships.
- ›Computer graphics: Rotating points, camera matrices, and animation all rely on the rotation formula (x′, y′) = (x cos θ − y sin θ, x sin θ + y cos θ).
- ›Electrical engineering: AC voltage and current are modelled as V = V₀ sin(ωt + φ), a direct application of the unit circle with time as the angle parameter.
- ›Navigation and GPS: Haversine and great-circle distance formulas use spherical trigonometry — an extension of the unit circle to 3D.
Frequently Asked Questions
Why is the radius of the unit circle exactly 1?
Setting r = 1 is a deliberate definition chosen for mathematical elegance:
- ›With r = 1, the point (x, y) on the circle directly gives (cos θ, sin θ) — no scaling factor.
- ›A circle of radius r would require x/r = cos θ. Dividing by 1 is trivially simple.
- ›The Pythagorean identity sin²θ + cos²θ = 1 follows directly from x² + y² = r² when r = 1.
- ›The unit circle is the natural domain for defining trig functions for all real angles, not just acute angles in triangles.
How do I read coordinates from the unit circle?
To read a point from the unit circle:
- ›Locate the angle θ measured counterclockwise from the positive x-axis (3 o'clock position).
- ›The horizontal distance from the origin to the point is cos θ (positive right, negative left).
- ›The vertical distance from the origin to the point is sin θ (positive up, negative down).
- ›At θ = 45°, the point is (√2/2, √2/2) — the same horizontal and vertical distance.
- ›At θ = 180°, the point is (−1, 0) — fully to the left on the x-axis.
Which angles have exact trig values?
Exact radical values exist for all multiples of 30° and 45°:
- ›0°, 90°, 180°, 270°, 360°: values are 0, ±1 — from the axes.
- ›30°, 150°, 210°, 330° (π/6 multiples): sin = ±1/2, cos = ±√3/2.
- ›45°, 135°, 225°, 315° (π/4 multiples): sin = cos = ±√2/2.
- ›60°, 120°, 240°, 300° (π/3 multiples): sin = ±√3/2, cos = ±1/2.
- ›For arbitrary angles like 37°, only decimal approximations are possible.
What is the ASTC rule (All Students Take Calculus)?
ASTC — starting from Quadrant I and going counterclockwise:
- ›Q I (0°–90°) — All: sin, cos, tan, csc, sec, cot are all positive.
- ›Q II (90°–180°) — Sine: sin and its reciprocal csc are positive; all others negative.
- ›Q III (180°–270°) — Tangent: tan and its reciprocal cot are positive; sin, cos, csc, sec negative.
- ›Q IV (270°–360°) — Cosine: cos and its reciprocal sec are positive; all others negative.
Knowing ASTC means you never need to memorise every value — just find the reference angle and apply the right sign.
Why is tan(90°) undefined?
Functions become undefined when their denominator is zero:
- ›tan θ = sin θ / cos θ → undefined when cos θ = 0, i.e. θ = 90°, 270°, and all odd multiples of 90°.
- ›sec θ = 1 / cos θ → undefined at the same angles as tan.
- ›cot θ = cos θ / sin θ → undefined when sin θ = 0, i.e. θ = 0°, 180°, 360°, …
- ›csc θ = 1 / sin θ → undefined at the same angles as cot.
The calculator displays "Undefined" for these cases rather than showing a numerical error.
What is a reference angle and how is it calculated?
Reference angle formulas (for θ in the range 0° to 360°):
- ›Quadrant I (0°–90°): reference angle = θ
- ›Quadrant II (90°–180°): reference angle = 180° − θ
- ›Quadrant III (180°–270°): reference angle = θ − 180°
- ›Quadrant IV (270°–360°): reference angle = 360° − θ
For angles outside 0°–360°, reduce to the coterminal angle first by adding or subtracting multiples of 360°.
Why does trigonometry matter outside of geometry class?
Trigonometry from the unit circle underlies almost every branch of STEM:
- ›Fourier analysis: any periodic signal = sum of sin and cos waves at different frequencies.
- ›Euler's formula: e^(iθ) = cos θ + i sin θ — connects complex exponentials to the unit circle.
- ›Physics: simple harmonic motion x(t) = A cos(ωt + φ) uses unit-circle coordinates directly.
- ›Computer graphics: 2D rotation matrices R(θ) use cos and sin to rotate every pixel.
- ›Electrical engineering: impedance, phase angles, and phasors all use complex trig.