Cosine Calculator
Calculate the cosine of any angle, find the angle from a cosine value with inverse cosine, or solve a triangle side or angle using the law of cosines. Choose your mode, enter the values, and get the exact answer with step-by-step work.
Formula
Worked example
For θ = 60°: convert to radians (60 × π/180 = π/3 ≈ 1.0472 rad), then cos(π/3) = 0.5. sin(60°) = √3/2 ≈ 0.8660, tan(60°) = √3 ≈ 1.7321. General solution: θ = ±60° + 360°k. Law of cosines check: if a=6, b=5, c=7, cos(A) = (25+49-36)/(70) = 38/70 ≈ 0.5429, A ≈ 57.12°.
What cosine measures and how this calculator works
In a right triangle the cosine of an acute angle is the length of the adjacent side divided by the hypotenuse. The unit circle extends this to every angle: draw a radius of length one at angle θ from the positive x-axis, and the cosine is the x-coordinate of the tip. This tool has four modes. Cosine mode computes cos(θ) for any angle and also shows sin(θ), tan(θ), the quadrant, and the general solution for all angles with the same cosine. Inverse cosine mode takes a value between -1 and 1 and returns the principal angle plus every angle in the infinite solution set. The two law-of-cosines modes solve for either a missing side or a missing angle in any triangle, not just right triangles.
Angle units: degrees, radians, milliradians, and pi radians
Degrees divide a full circle into 360 equal parts and are the everyday standard. Radians are the natural unit for calculus and physics: one radian is the angle that subtends an arc equal to the radius, and a full circle is 2π radians. Milliradians (mrad) are used in ballistics and optics, where the full circle is 2000π mrad. Pi radians express the angle as a multiple of π, so 1 pirad = 180°. To convert degrees to radians multiply by π/180; to get mrad multiply the radian value by 1000. This calculator handles all four units so you never have to convert manually before entering your angle.
Inverse cosine and the complete solution set
The inverse cosine function arccos(x) returns the principal value, the unique angle between 0° and 180° whose cosine is x. But cosine is periodic with period 360° and even (cos(-θ) = cos(θ)), so every cosine value corresponds to infinitely many angles. If the principal value is α, then both α and 360° - α have the same cosine, and adding 360° to either gives another solution. The complete set is θ = ±α + 360°k for any integer k. The calculator shows all of this in the general-solution and all-solutions fields so you can pick the angle your problem actually needs.
Law of cosines: solve any triangle
The law of cosines states a² = b² + c² - 2bc·cos(A), where a is the side opposite angle A, and b and c are the other two sides. It reduces to the Pythagorean theorem when A = 90°, because cos(90°) = 0. Use the find-a-side mode when you know two sides and the angle between them (SAS). Use the find-an-angle mode when you know all three sides (SSS). To solve a triangle completely, find one unknown at a time and reuse the result as a known value in the next step. The sign of cos(A) tells you immediately whether the angle is acute (positive) or obtuse (negative), which is useful when checking your triangle makes geometric sense.
Sign, symmetry, and periodicity
Cosine is positive in quadrants I and IV, where the terminal ray points to the right of the y-axis, and negative in quadrants II and III, where it points to the left. It is an even function, meaning cos(-θ) = cos(θ), so reflecting the angle across the x-axis does not change the cosine. The period is 360° (or 2π radians): cos(θ + 360°) = cos(θ) for every θ. These two properties together mean you only need to know cosine values for 0° to 180° to reconstruct the function everywhere: for angles in the third or fourth quadrant, reflect or shift to find the equivalent first- or second-quadrant angle.
Cosine values for common angles (0° to 360°)
| Angle (degrees) | Angle (radians) | Exact cos θ | Decimal | Quadrant |
|---|---|---|---|---|
| 0° | 0 | 1 | 1.0000 | Axis |
| 30° | π/6 | √3/2 | 0.8660 | I |
| 45° | π/4 | √2/2 | 0.7071 | I |
| 60° | π/3 | 1/2 | 0.5000 | I |
| 90° | π/2 | 0 | 0.0000 | Axis |
| 120° | 2π/3 | -1/2 | -0.5000 | II |
| 135° | 3π/4 | -√2/2 | -0.7071 | II |
| 150° | 5π/6 | -√3/2 | -0.8660 | II |
| 180° | π | -1 | -1.0000 | Axis |
| 210° | 7π/6 | -√3/2 | -0.8660 | III |
| 225° | 5π/4 | -√2/2 | -0.7071 | III |
| 240° | 4π/3 | -1/2 | -0.5000 | III |
| 270° | 3π/2 | 0 | 0.0000 | Axis |
| 300° | 5π/3 | 1/2 | 0.5000 | IV |
| 315° | 7π/4 | √2/2 | 0.7071 | IV |
| 330° | 11π/6 | √3/2 | 0.8660 | IV |
| 360° | 2π | 1 | 1.0000 | Axis |
Exact and decimal cosine values covering a full circle. Sign flips at 90° and 270°.
Frequently asked questions
What is the range of the cosine function?
Cosine always returns a value between -1 and 1 inclusive. It reaches its maximum of 1 at 0° (and every multiple of 360°), drops to 0 at 90°, falls to its minimum of -1 at 180°, rises back through 0 at 270°, and returns to 1 at 360°. No real angle can produce a cosine outside this band.
How do I convert degrees to radians?
Multiply the degree value by π/180. For example, 60° becomes 60 × π/180 = π/3 ≈ 1.0472 radians. To go the other way, multiply radians by 180/π. This calculator accepts degrees, radians, milliradians, and pi radians directly, so you can skip manual conversion.
What does the inverse cosine (arccos) calculator do?
Arccos takes a cosine value between -1 and 1 and returns the angle whose cosine it is. The principal value is always in the range 0° to 180°. Because cosine repeats every 360° and is symmetric about the x-axis, there are infinitely many angles with the same cosine: the general solution is θ = ±arccos(x) + 360°k for any integer k. The calculator shows the principal angle, its radian equivalent, and the complete solution set.
When do I use the law of cosines instead of basic trig?
Use the law of cosines when your triangle is not a right triangle and you know either two sides and the included angle (SAS) or all three sides (SSS). For right triangles the Pythagorean theorem and basic sin/cos/tan ratios are simpler. The law of cosines is a generalisation that works for any triangle, reducing to a² = b² + c² whenever the angle is exactly 90°.
Why does cos(-θ) equal cos(θ)?
Cosine is an even function. On the unit circle, an angle -θ is the reflection of θ across the x-axis. The reflection changes the y-coordinate (which is the sine) but leaves the x-coordinate (which is the cosine) unchanged. So cosine is identical for an angle and its negative.
What are milliradians and when are they used?
A milliradian (mrad) is one-thousandth of a radian. The full circle contains 2000π mrad (roughly 6283 mrad). Milliradians are common in military ballistics and precision optics because at long ranges 1 mrad subtends approximately 1 metre per 1000 metres of distance, making range corrections straightforward. This calculator accepts mrad as an input unit directly.