Choose a calculation mode
Calculation history
Saved entries remain in this browser. They are not sent to a database.
Features included in this calculator
- All six trigonometric functions
- Exact radical simplification
- Degree and radian conversion
- Rational multiples of π
- Negative and coterminal angles
- Quadrant and axis detection
- Reference-angle calculation
- Undefined-value warnings
- Rationalized reciprocals
- Unit-circle coordinates
- Exact expression arithmetic
- Batch line evaluation
- Special-angle table builder
- Inverse exact-value lookup
- Known-ratio reconstruction
- Coordinate-based evaluation
- Right-triangle calculations
- Identity checking
- Random practice questions
- CSV and PDF exports
- Copy, print, and sharing tools
- Local calculation history
- Responsive mathematical keyboard
- Light and dark themes
How to use the calculator
Choose the correct mode
Use All six functions when you need a complete unit-circle analysis. Select Single function for one concise result. Choose Exact expression when several trigonometric values must be combined and simplified.
Enter the angle
Type an angle such as 30°, -225°, pi/6, 5pi/4, or 17pi/3. A degree symbol overrides the default unit. Any input containing pi is interpreted as a radian expression.
Control the output
Choose decimal precision and standard or scientific notation. The symbolic exact value remains visible whenever the angle belongs to the supported special-angle set. Decimal settings only affect approximations.
Export or store the result
Use CSV for spreadsheet-friendly tables. Use PDF for a compact printable record. You may also copy the result, print the page, share supported browser links, or store an entry in local history.
Formulas used
Right-triangle ratios
Reciprocal identities
Pythagorean identities
Degree and radian conversion
Reference-angle rules
How symbolic values remain exact
This implementation does not merely store formatted answer strings. It represents every supported exact value inside the number field generated by √2 and √3. A value is stored as four rational coefficients:
Addition combines matching coefficients. Multiplication uses the relations \((\sqrt2)^2=2\), \((\sqrt3)^2=3\), and \(\sqrt2\sqrt3=\sqrt6\). Division solves an exact rational linear system. Therefore reciprocal values such as \(1/(\sqrt3/2)\) become \(2\sqrt3/3\) without floating-point guessing.
The supported exact grid uses increments of fifteen degrees. It contains the standard unit-circle angles and additional angles such as 15°, 75°, 105°, and 345°. Their values follow from sum and difference identities.
Common exact values
| Angle | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 15° | π/12 | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 75° | 5π/12 | (√6 + √2)/4 | (√6 − √2)/4 | 2 + √3 |
| 90° | π/2 | 1 | 0 | Undefined |
Understanding exact trigonometric values
Exact trigonometric values describe angles without rounding. They use integers, fractions, radicals, and π. An answer such as √3/2 preserves the complete mathematical value. A decimal like 0.8660254038 is only an approximation.
The unit circle gives a direct geometric interpretation. Its radius is one. A terminal point has coordinates \((\cos\theta,\sin\theta)\). This makes cosine the horizontal coordinate and sine the vertical coordinate. Tangent is their quotient when cosine is not zero.
Reference angles simplify evaluation. A reference angle is the acute angle formed with the x-axis. Angles sharing a reference angle have the same unsigned basic values. Their quadrants determine the signs. Sine is positive in Quadrants I and II. Cosine is positive in Quadrants I and IV. Tangent is positive in Quadrants I and III.
Coterminal angles end at the same terminal side. Adding or subtracting 360° does not change any trigonometric value. In radians, adding or subtracting 2π has the same effect. The calculator normalizes large or negative angles before evaluation.
Undefined values occur through division by zero. Tangent and secant are undefined when cosine equals zero. Cotangent and cosecant are undefined when sine equals zero. The result panel identifies these conditions rather than displaying a misleading large decimal.
Reciprocal values may initially contain radicals in denominators. Rationalization rewrites the expression with a rational denominator. For example, the reciprocal of √3 is √3/3. Exact algebra maintains equality while producing a conventional final form.
Fifteen-degree angles come from angle-sum and angle-difference identities. They expand the familiar 30°, 45°, and 60° set. Values at 15° and 75° involve both √2 and √6. Symmetry then supplies the matching values in every quadrant.
Expression mode combines these values symbolically. It can verify numerical instances of identities and simplify sums, products, quotients, and integer powers. Batch mode applies the same engine to several independent lines. Table mode organizes selected functions across many angles.
Examples
Find sin(150°)
The angle lies in Quadrant II. Its reference angle is 30°. Sine is positive in Quadrant II. Therefore \(\sin150^\circ=\sin30^\circ=1/2\).
Find sec(120°)
The reference angle is 60°. Cosine is negative in Quadrant II, so \(\cos120^\circ=-1/2\). Secant is the reciprocal. Therefore \(\sec120^\circ=-2\).
Find tan(315°)
The reference angle is 45°. Tangent is negative in Quadrant IV. Since \(\tan45^\circ=1\), the exact result is \(-1\).
Evaluate sin²(45°) + cos²(45°)
Both sine and cosine equal √2/2. Squaring each gives 1/2. Their sum is one, confirming the Pythagorean identity at 45°.
Frequently asked questions
What makes a trigonometric value exact?
An exact value uses fractions, radicals, integers, or π without rounding. It preserves the full mathematical quantity rather than a finite decimal approximation.
Which angles receive symbolic answers?
The built-in symbolic engine supports every angle equivalent to a multiple of 15°. Other valid angles receive decimal approximations and complete angle analysis.
Can I enter negative angles?
Yes. Negative angles are normalized to an equivalent angle between 0° and 360°. The original and coterminal forms remain visible in the result.
Can I use radians?
Yes. Enter rational multiples such as pi/6, 5pi/4, or -7pi/3. Plain numeric radians are also accepted for decimal evaluation.
Why is tangent sometimes undefined?
Tangent equals sine divided by cosine. It is undefined whenever cosine is zero, including 90° and 270° plus complete rotations.
Does the identity checker prove identities?
No. It verifies equality at one selected special angle. A universal proof requires algebraic transformation valid throughout the common domain.
Where is calculation history stored?
History uses local browser storage. It stays on the current device and browser profile until you remove it or clear site data.
Can I export a table?
Yes. Submit the table mode with the CSV button. The downloaded file contains headings and all calculated rows for spreadsheet use.