Calculator inputs
Formula used
Angles are converted to radians before PHP evaluates the sine function. Standard unit-circle angles are also checked for exact fractions and radical expressions.
How to use this calculator
- Choose angle mode or direct sine-value mode.
- Enter an angle, decimal, or supported fraction.
- Select degrees, radians, or gradians when using angle mode.
- Set the maximum denominator, precision, and tolerance.
- Select your preferred exact or approximate output.
- Press Calculate to view the fraction, errors, and trigonometric details.
- Use Copy, CSV, or Save PDF to keep the result.
For radian inputs, expressions such as pi, pi/6, -pi/2, and 2*pi/3 are supported.
Exact sine values
| Angle | Radians | Sine decimal | Exact form |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | 0.5 | 1/2 |
| 45° | π/4 | 0.707106... | √2/2 |
| 60° | π/3 | 0.866025... | √3/2 |
| 90° | π/2 | 1 | 1 |
| 210° | 7π/6 | -0.5 | -1/2 |
Recent calculation history
| Input | Sine | Fraction | Error |
|---|---|---|---|
| No saved calculations yet. | |||
Understanding sine fractions
Sine compares the opposite side of a right triangle with its hypotenuse. Its value always stays between negative one and positive one. The same relationship also appears on the unit circle.
Some angles produce exact rational values, including zero, one-half, and one. Other familiar angles produce exact radical forms such as √2/2. Those radical expressions are exact even though their decimals continue indefinitely.
A fraction approximation replaces a decimal with a nearby ratio of integers. Increasing the maximum denominator can usually improve accuracy. It may also produce a fraction that is harder to read.
The tolerance controls how close the fraction should be before the search stops. A smaller tolerance requests a closer numerical match. The denominator limit still prevents excessively large results.
Negative angles and angles beyond one full turn are supported. The calculator normalizes angles to identify quadrants and reference angles. Coterminal angles have the same sine value.
Direct-value mode is useful when another calculation already produced a sine decimal. It also accepts simple fractions such as 1/2. Values outside the interval from -1 to 1 are rejected.
Frequently asked questions
Can every sine value be written as an exact fraction?
No. Many sine values are irrational, so the calculator provides an exact radical when recognized or a close rational approximation.
Why does sin(45°) show √2/2?
That radical is the exact value. A decimal or ordinary fraction can only approximate it.
What does maximum denominator mean?
It sets the largest allowed denominator in the approximated fraction. Larger limits can improve accuracy but create longer fractions.
Can I enter radians using pi?
Yes. Select radians and enter forms such as pi/6, -pi/2, or 2*pi/3.
Can I enter a fraction as the sine value?
Yes. Direct-value mode accepts entries such as 1/2 or -3/4, provided the result remains between -1 and 1.
Why is tangent sometimes undefined?
Tangent equals sine divided by cosine. It is undefined when cosine is zero, including angles such as 90° and 270°.
What is a reference angle?
It is the positive acute angle between the terminal side and the x-axis. It helps identify related trigonometric values.
How is percentage error calculated?
The absolute approximation error is divided by the absolute sine value and multiplied by 100. Zero values are handled separately.
Does the PDF button create a file automatically?
It opens the browser print dialog. Choose the browser option for saving the page as a PDF.