# What is the cos of my angle?

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

- Page: https://www.acalculator.org/math/cosine-calculator
- JSON spec: https://www.acalculator.org/math/cosine-calculator.json
- Version: 83c9f8d4b79a

## Default answer

Example with the default inputs (Angle θ 30 °): cos(30°) = 0.866025.

## Inputs

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

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| cos | cos θ | Cosine of θ: the x 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°. |
| sec | sec θ | 1 ÷ cos θ; empty where cos 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

cos θ is the x 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 cos θ is computed in 64-bit floating point. sec θ = 1 ÷ cos 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 cos 90° is exactly 0.
- sec θ shows nothing where cos θ = 0 (90°, 270°, …).

## Worked examples

1. angle = 1.047198 gives cos = 0.5, exact = cos 60° = 1/2, sec = 2, reference = 1.047198, quadrant = Quadrant I. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: cos 60° = 1/2, cos 30° = √3/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
2. angle = 2.617994 gives cos = -0.866025, exact = cos 150° = -√3/2, quadrant = Quadrant II, reference = 0.523599. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: cos 60° = 1/2, cos 30° = √3/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
3. angle = 1.570796 gives cos = 0, exact = cos 90° = 0, quadrant = On the positive y-axis. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: cos 60° = 1/2, cos 30° = √3/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle.
4. angle = 1 gives cos = 0.540302, sec = 1.850816. Source: NIST DLMF §4.14 (cosine of a real angle), https://dlmf.nist.gov/4.14.
5. angle = 3.490659 gives cos = -0.939693, quadrant = Quadrant III. Source: NIST DLMF §4.14, https://dlmf.nist.gov/4.14.

## FAQ

### What is the cosine 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 cos θ is its x value. In a right triangle with acute angle θ, cos θ = adjacent side ÷ hypotenuse.

### What is cos 60°?

cos 60° = 1/2 = 0.5 exactly. The other common values are cos 30° = √3/2 ≈ 0.8660, cos 45° = √2/2 ≈ 0.7071 and cos 90° = 0.

### Should I use degrees or radians?

Use the unit the angle is given in. A full turn is 360° or 2π radians. Typing 60 with the unit set to radians gives cos 60 rad ≈ −0.9524, not 0.5.

### When is cosine negative?

cos θ is negative when the point on the unit circle is left of the y-axis: for angles between 90° and 270° (quadrants II and III). cos 150° = −√3/2.

### Is cos(−θ) the same as cos θ?

Yes. Cosine is an even function: turning the same amount clockwise lands at the same x value. cos(−60°) = cos 60° = 1/2.

### What is sec θ?

The secant, sec θ = 1 ÷ cos θ. It is undefined where cos θ = 0 (90°, 270°), so the calculator leaves it empty there.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (cosine as the x 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 (sec θ = 1 ÷ cos 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)