# What is the sin of my angle?

Finds sin θ of any angle in degrees or radians, with the exact value at multiples of 15°, csc θ, and the reference angle.

- Page: https://www.acalculator.org/math/sine-calculator
- JSON spec: https://www.acalculator.org/math/sine-calculator.json
- Version: 60ededd41fc1

## Default answer

Example with the default inputs (Angle θ 30 °): sin(30°) = 0.5.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| angle | Angle θ | Any angle, in degrees or radians; a negative angle turns clockwise. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| sin | sin θ | Sine of θ: the y value of the point at angle θ on the unit circle. |
| exact | Exact value | The value as a fraction and square roots, for angles that are a multiple of 15°. |
| csc | csc θ | 1 ÷ sin θ; empty where sin θ = 0. |
| reference | Reference angle | The acute angle between θ and the x-axis, from 0° to 90°. |
| quadrant | Quadrant | Where θ ends on the unit circle. |

## Method

sin θ is the y value of the point at angle θ on the unit circle. At a multiple of 15° the exact value comes from the special angles and the quadrant sign; otherwise sin θ is computed in 64-bit floating point. csc θ = 1 ÷ sin θ.

## Assumptions

- The angle can be typed in degrees or radians, from −1,000,000 to 1,000,000 radians, and can be negative or more than 360°.
- An angle within one part in 10¹² of a multiple of 15° counts as that multiple, so sin 180° is exactly 0.
- csc θ shows nothing where sin θ = 0 (0°, 180°, 360°, …).

## Worked examples

1. angle = 0.523599 gives sin = 0.5, exact = sin 30° = 1/2, csc = 2, reference = 0.523599, quadrant = Quadrant I. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
2. angle = 3.926991 gives sin = -0.707107, exact = sin 225° = -√2/2, quadrant = Quadrant III, reference = 0.785398. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
3. angle = 3.141593 gives sin = 0, exact = sin 180° = 0, quadrant = On the negative x-axis. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
4. angle = 1 gives sin = 0.841471, csc = 1.188395. Source: NIST DLMF §4.14 (sine of a real angle), https://dlmf.nist.gov/4.14.
5. angle = 3.490659 gives sin = -0.34202, quadrant = Quadrant III. Source: NIST DLMF §4.14, https://dlmf.nist.gov/4.14.

## FAQ

### What is the sine of an angle?

Draw the angle θ from the positive x-axis on a circle of radius 1 centred at the origin. The point where it meets the circle is (cos θ, sin θ), so sin θ is its y value. In a right triangle with acute angle θ, sin θ = opposite side ÷ hypotenuse.

### What is sin 30°?

sin 30° = 1/2 = 0.5 exactly. The other common values are sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660 and sin 90° = 1.

### Should I use degrees or radians?

Use the unit the angle is given in. A full turn is 360° or 2π radians, so 1 radian is about 57.2958°. Typing 30 with the unit set to radians gives sin 30 rad ≈ −0.9880, not 0.5.

### Can sine be negative?

Yes. sin θ is negative when the point on the unit circle is below the x-axis: for angles between 180° and 360° (quadrants III and IV). sin 225° = −√2/2.

### What is the largest value of sin θ?

sin θ is always between −1 and 1. It is 1 at 90° (plus any whole number of turns) and −1 at 270°.

### What is csc θ?

The cosecant, csc θ = 1 ÷ sin θ. It is undefined where sin θ = 0 (0°, 180°, 360°), so the calculator leaves it empty there.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (sine as the y value on the unit circle; special angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (csc θ = 1 ÷ sin θ). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions (retrieved 2026-10-01)
- NIST Digital Library of Mathematical Functions, §4.14 Definitions and Periodicity (trigonometric functions). https://dlmf.nist.gov/4.14 (retrieved 2026-10-01)