What is the cos of my angle?
Type an angle in degrees or radians. The cos calculator shows cos θ, its exact value when the angle is a multiple of 15°, sec θ, the reference angle and the quadrant.
- cos θ
- 0.866025
cos(30°) = 0.866025.
- Exact value
- cos 30° = √3/2
- sec θ
- 1.154701
- Reference angle
- 30°
- Quadrant
- Quadrant I
cos θ: 0.866025. cos(30°) = 0.866025.
How to calculate
Finds cos θ of any angle in degrees or radians, with the exact value at multiples of 15°, sec θ, and the reference angle.
Example with the default inputs (Angle θ 30 °): cos(30°) = 0.866025.
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.
- 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
Each example is checked against the calculator on every build.
- Angle θ 60° gives cos θ 0.5, Exact value cos 60° = 1/2, sec θ 2, Reference angle 60°, 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
- Angle θ 150° gives cos θ -0.866025, Exact value cos 150° = -√3/2, Quadrant Quadrant II, Reference angle 30°.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
- Angle θ 90° gives cos θ 0, Exact value 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
- Angle θ 57.3° gives cos θ 0.540302, sec θ 1.850816.Source: NIST DLMF §4.14 (cosine of a real angle), https://dlmf.nist.gov/4.14
- Angle θ 200° gives cos θ -0.939693, Quadrant Quadrant III.Source: NIST DLMF §4.14, https://dlmf.nist.gov/4.14
How it works
Draw the angle θ from the positive x-axis, turning counterclockwise for a positive angle. It meets the unit circle (radius 1) at the point (cos θ, sin θ). The cosine is the x value:
- cos θ = x, and in a right triangle, cos θ = adjacent ÷ hypotenuse.
- sec θ = 1 ÷ cos θ, undefined where cos θ = 0.
- Reference angle: the acute angle between θ and the x-axis, from 0° to 90°.
- Quadrant: I (0° to 90°), II (90° to 180°), III (180° to 270°) or IV (270° to 360°), after whole turns are taken off. An angle that ends on an axis shows the axis instead.
Exact values. When θ is a multiple of 15° (π/12 radians), the calculator shows the exact value. It takes the reference angle's cosine from this table and gives it the sign of the quadrant (positive in I and IV, negative in II and III):
| Reference angle | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
|---|---|---|---|---|---|---|---|
| cos | 1 | (√6 + √2)/4 | √3/2 | √2/2 | 1/2 | (√6 − √2)/4 | 0 |
An angle within one part in 10¹² (relative, and at least 10⁻¹² radians) of a multiple of 15° counts as that multiple. So cos 90° is exactly 0, not the 6.1 × 10⁻¹⁷ that a rounded π/2 gives. For any other angle, cos θ is computed in 64-bit floating point.
Output format. cos θ and sec θ show as decimal numbers. The exact value shows as text, such as cos 60° = 1/2 (- for a negative value). The reference angle shows in degrees.
Assumptions
- The angle is typed in degrees or radians, from −1,000,000 to 1,000,000 radians. Negative angles turn clockwise.
- sec θ shows nothing at 90°, 270° and the other angles where cos θ = 0.
Worked examples by hand
θ = 60°. 60° is in quadrant I, so cos 60° = 1/2 = 0.5 and sec 60° = 2. The reference angle is 60°.
θ = 150°. 180° − 150° = 30°, so the reference angle is 30° and the angle is in quadrant II, where the cosine is negative: cos 150° = −√3/2 ≈ −0.8660254037844386.
θ = 90°. The point is (0, 1), so cos 90° = 0, on the positive y-axis. sec 90° is undefined.
θ = 1 radian. cos 1 ≈ 0.5403023058681398, and sec 1 = 1 ÷ 0.5403023058681398 ≈ 1.8508157176809255.
θ = 200°. 200° is in quadrant III (reference angle 20°), so the cosine is negative: cos 200° ≈ −0.9396926207859084.
Other questions people ask
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.