🔺 Trigonometry Calculator
Find sin, cos, tan, csc, sec and cot for any angle in degrees or radians. Includes a visual unit circle and special-angle reference table.
What is this tool?
Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, and the periodic functions that arise from those relationships. The three primary trigonometric functions — sine (sin), cosine (cos) and tangent (tan) — are defined from the ratios of sides in a right triangle: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. Their reciprocals are cosecant (csc), secant (sec) and cotangent (cot).
This calculator takes any angle — positive or negative, in degrees or radians — and instantly returns all six function values to eight decimal places. It handles special angles like 30°, 45° and 60° exactly, works with negative angles using symmetry rules, and warns you when a value is undefined (for example, tan 90°).
A visual unit circle shows how each angle relates to its sine and cosine coordinates on the circle. Everything runs locally in your browser with no registration, so it is ideal for homework, exam practice, physics problems, engineering sketches and game development calculations.
For related calculations, try the triangle calculator, the Pythagorean theorem calculator and the vector calculator to explore more useful tools.
How it works
The calculator converts your angle to radians internally (degrees × π ÷ 180), then evaluates the six functions: sin θ, cos θ, tan θ = sin θ ÷ cos θ, csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, and cot θ = 1 ÷ tan θ (or cos θ ÷ sin θ).
Special cases are handled explicitly. When cos θ = 0 (at 90° and 270°), tan θ and sec θ are undefined — the tool reports "undefined" instead of returning an infinite value. Similarly, when sin θ = 0 (at 0° and 180°), csc θ and cot θ are undefined. Negative angles are evaluated using the symmetry identities: sin(−θ) = −sin θ, cos(−θ) = cos θ, and tan(−θ) = −tan θ.
A useful derived value is the reciprocal relationship check: csc θ = 1 ÷ sin θ must hold exactly, so you can verify any single function value by computing its reciprocal. The unit circle diagram plots the point (cos θ, sin θ) for your angle, showing why cos is the x-coordinate and sin is the y-coordinate.Unit Circle Visual
Each angle maps to a point (cos θ, sin θ) on the unit circle.
Special Angle Values Reference Table
Exact trigonometric values for the most commonly used special angles.
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | undefined | 1 | undefined |
| 30° | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | 1 | 0 | undefined | 1 | undefined | 0 |
Trigonometry Identities Quick Reference
Essential identities used to simplify trigonometric expressions.
| Identity | Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Tangent | tan θ = sin θ ÷ cos θ |
| Reciprocal | csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ |
| Negative angle | sin(−θ) = −sin θ, cos(−θ) = cos θ |
| Double angle | sin 2θ = 2 sin θ cos θ |
| Co-function | sin(90° − θ) = cos θ |
How to use
- Enter the angle value in the input box.
- Choose the unit — degrees or radians.
- Press Calculate to evaluate all six trigonometric functions.
- Read the results; undefined values are shown where the function has no value.
- Switch the unit and recalculate to see the same angle expressed differently.
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees divide a full circle into 360 parts, while radians measure angles by arc length, with a full circle equal to 2π radians. To convert, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
Why does the calculator show "undefined" for tan 90°?
Tangent is defined as sin θ ÷ cos θ. At 90°, cos 90° = 0, and dividing by zero has no defined value. The graph of tan θ has a vertical asymptote at 90° and 270° for this reason.
Can I enter negative angles?
Yes. Negative angles are measured clockwise from the positive x-axis and are evaluated using the symmetry identities: sin(−θ) = −sin θ, cos(−θ) = cos θ and tan(−θ) = −tan θ.
How do the special angles work?
The special angles 30°, 45° and 60° have exact trigonometric values involving 1/2, √2/2 and √3/2. The calculator returns them as precise decimals, and the reference table on this page lists the exact forms.
What is the unit circle?
The unit circle is a circle of radius 1 centred at the origin. For any angle θ, the point where the radius meets the circle has coordinates (cos θ, sin θ), which is why cosine is the x-coordinate and sine is the y-coordinate.
Which functions are reciprocals of which?
Cosecant is the reciprocal of sine (csc θ = 1/sin θ), secant is the reciprocal of cosine (sec θ = 1/cos θ), and cotangent is the reciprocal of tangent (cot θ = 1/tan θ).
Tips & Advice
Memorise the special angles: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2 — the cosines are the same values in reverse order. For quick mental checks, remember that sin and cos always stay between −1 and 1, while tan can grow without bound. When working in degrees, make sure your calculator is set to degrees mode — mixing degrees and radians is the most common source of wrong answers in trigonometry homework. Use the reciprocal functions to cross-check: if your calculator says sin 30° = 0.5, then csc 30° must be exactly 2. For physics problems, radians are usually preferred, so convert early and work consistently in one unit throughout.
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Sources & References
- NIST. Digital Library of Mathematical Functions - Trigonometric Functions. dlmf.nist.gov.
- Wolfram MathWorld. Trigonometric Functions. mathworld.wolfram.com.
Last reviewed: August 2026.
Limitations
This tool evaluates standard trigonometric functions using JavaScript floating-point arithmetic. Results are rounded to eight decimal places.
What this tool does not account for:
- Floating-point precision: for extremely large angles, repeated conversion to radians can introduce small rounding errors.
- Undefined values: functions like tan 90° are mathematically undefined; the tool displays "undefined" rather than a numerical result.
- Non-right-triangle contexts: these functions also describe periodic wave behaviour in physics and engineering, which this tool does not visualise beyond the unit circle.
For exam and homework answers, always verify results with the exact special-angle values where applicable.