Cosine Calculator
Cosine should be the simplest lookup in trigonometry, yet it generates a steady stream of wrong answers, nearly all from one cause: the angle was in degrees and the tool was thinking in radians, or the reverse. A dedicated calculator with an explicit unit selector removes that failure mode.
This calculator takes an angle in degrees or radians and returns cos(x) to ten decimal places, together with your angle converted into radians so you can see exactly what was evaluated. No inverse mode, no triangle solver bolted on: the forward function, done carefully.
Forward cosine only: angle in, cos(x) out. For the inverse (a cosine value back to an angle), that is arccos, a separate tool.
Quick Answer
Key takeaways
- Cosine is the adjacent-over-hypotenuse ratio and the x-coordinate on the unit circle; its value is always between -1 and 1.
- The unit selector is the result. Degrees and radians readings of the same number differ wildly: cos(60 degrees) = 0.5, cos(60 radians) = -0.9524.
- Cosine is even and periodic: cos(-x) = cos(x) and cos(x + 360 degrees) = cos(x), so negative and oversized angles are safe to enter.
- This tool is forward-only by design. Its reciprocal-family sibling returns 1/sin(x) instead: cosecant, not cosine.
- Quick Answer
- How to use the cosine calculator
- The cosine formula: adjacent over hypotenuse, x on the unit circle
- Worked example, checked against this exact tool: cos(60 degrees)
- Degrees versus radians: the error that causes all the others
- Where cosine earns its keep
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the cosine calculator
Enter the angle, set the unit to match the problem statement, and press Calculate Cosine. Read two outputs: the cosine itself, to ten decimal places, and the angle in radians. That second number is the audit trail: it shows the exact radian figure the cosine was taken of, so if the answer looks wrong you can see immediately whether the conversion or the function is at fault.
Know what this page is not. It does not run arccos; if you hold a cosine value and need the angle back, the acos calculator is the inverse tool, and the arctan calculator does the same job for tangent values. And it is not the reciprocal of sine: that function is cosecant, covered by its sibling page, the cosecant calculator, which returns csc(x) = 1/sin(x) with a full six-function table. Cosine stands on its own as adjacent over hypotenuse, not as anyone’s reciprocal.
The cosine formula: adjacent over hypotenuse, x on the unit circle
Two definitions, one function, and fluency means holding both.
The NIST Digital Library of Mathematical Functions builds the theory on the unit-circle definition, which is the one that survives contact with negative angles, angles past 360 degrees, and calculus. The triangle definition is the special case for acute angles: at 60 degrees the adjacent side of a 30-60-90 triangle is half the hypotenuse, which is why cos(60 degrees) is exactly 1/2. OpenStax Precalculus uses the same pair of definitions and the same cofunction identity worth memorizing: cos(x) = sin(90 degrees – x), so every cosine value is a sine value of the complementary angle.
Worked example, checked against this exact tool: cos(60 degrees)
The calculator is pre-filled with 60 degrees, and both figures below are what it returns.
| Output | Value | Note |
|---|---|---|
| Cosine value | 0.5000000000 | Exact value 1/2 |
| Angle in radians | 1.0471975512 | pi/3, shown to ten decimals |
Ten decimal places look like overkill until you are chaining results: a cosine rounded to 0.5 at the first step of a surveying or signal calculation carries its rounding error into every step after it. The reference values below are the ones this tool reproduces, and they are worth recognizing on sight, because a calculator answer that disagrees with them is a unit error until proven otherwise.
| Angle | Radians | cos |
|---|---|---|
| 0 degrees | 0 | 1.000000 |
| 30 degrees | pi/6 | 0.866025 |
| 45 degrees | pi/4 | 0.707107 |
| 60 degrees | pi/3 | 0.500000 |
| 90 degrees | pi/2 | 0.000000 |
| 120 degrees | 2pi/3 | -0.500000 |
| 180 degrees | pi | -1.000000 |
| 270 degrees | 3pi/2 | 0.000000 |
| 360 degrees | 2pi | 1.000000 |
Degrees versus radians: the error that causes all the others
Every wrong cosine story starts the same way. A student, a spreadsheet, or a line of code evaluates the cosine of a bare number, and the tool silently assumes a unit the human did not mean. Excel’s COS, JavaScript’s Math.cos, and Python’s math.cos all expect radians with no warning on the tin. The result for 60 is -0.9524, a plausible-looking number with no relationship to the intended 0.5, and plausibility is what makes the error dangerous: nothing about -0.9524 announces that it is wrong.
The defenses are habits, not talent. Check the unit before the arithmetic, every time. Sanity-check against the landmark values: a cosine outside -1 to 1 is impossible, and an acute angle must return a positive cosine under 1. And when an angle is given relative to a center, as in arc and sector work, confirm what the angle measures first; the central angle calculator handles that geometry when the angle itself is the unknown.
Where cosine earns its keep
Cosine is the projection function: it answers how much of something points in a given direction. The horizontal component of a force, the real power in an AC circuit (voltage times current times cos of the phase angle), the dot product between two directions in graphics and robotics, and the daily and annual cycles modeled with cosine waves all reduce to this one evaluation. The law of cosines extends it from right triangles to all triangles, c² = a² + b² – 2ab cos(C), and reduces to the Pythagorean theorem when C is 90 degrees, because cos(90 degrees) = 0 and the cross term vanishes.
Those applications are why precision and unit honesty matter more here than features. A phase angle read in the wrong unit does not produce a slightly wrong power figure; it produces a confidently wrong one. Run the angle, read the radian conversion back, and the number you carry into the bigger calculation is the number you meant.
What competing calculators miss
Cosine tools are everywhere, and most are fine as far as they go. The differences show up at the edges.
Miss 1: hidden unit assumptions. Tools that default to one unit with a faint toggle, or that accept a bare number with no unit shown in the result, manufacture the degrees-radians error instead of preventing it. Here the unit is a first-class input and the radian equivalent is printed in the result.
Miss 2: bundled modes that blur scope. Several top results merge cos, arccos, and sometimes triangle solving into one widget with cramped mode switches, and users routinely read an arccos output as a cosine. This page does one direction and labels the inverse tools as separate pages, which is the same separation your textbook uses.
Miss 3: overclaimed features in the surrounding text. Pages that promise inverse modes, calculation modes, and triangle solving the widget does not actually contain are worse than a short feature list: they teach the reader to expect outputs that never come. Everything described on this page is something the tool above demonstrably does.
One honest limit, repeated from the card: forward only. No arccos, no sides-to-angles triangle solving, no graph. Those are real needs, and they belong to their own tools rather than to a mode switch here.
Frequently Asked Questions
Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle, and the x-coordinate of the matching point on the unit circle. It is one of the core trigonometric functions: for any angle it returns a value between -1 and 1, with cos(0 degrees) = 1 and cos(180 degrees) = -1.
Exactly 0.5. This calculator shows it to ten decimal places as 0.5000000000, alongside the angle in radians, 1.0471975512. It is one of the standard exact values, along with cos(0 degrees) = 1, cos(45 degrees) = 0.7071067812, and cos(90 degrees) = 0.
0. The adjacent side shrinks to nothing as the angle opens to a right angle, and the unit-circle point sits at the top, where the x-coordinate is 0. The calculator displays 0.0000000000; internally the raw floating-point figure is a hair above zero, and the ten-decimal display rounds it to the exact answer.
Match the unit your problem is stated in, and set the selector to match. Degrees dominate school geometry and most field work; radians dominate calculus and programming libraries, where functions like Math.cos expect radians. Feeding 60 degrees into a radians calculation gives cos(60 rad) = -0.9524, a completely different answer, so the selector is the most important control on the page.
Cosine always lands between -1 and 1, inclusive. It is an even function, so cos(-x) = cos(x), and it repeats every 360 degrees (2pi radians): cos(420 degrees) = cos(60 degrees) = 0.5. Angles far outside one turn still work here, because the underlying math reduces them by the period automatically.
No. This tool runs the forward direction only: angle in, cosine out, plus the radian conversion of your angle. The inverse direction, cosine value in and angle out, is arccos, and it lives on its own page because the two directions answer different questions and mixing them in one form causes constant input errors.
The law of cosines, c² = a² + b² – 2ab cos(C), finds a side or an angle of any triangle, not just right ones. This calculator evaluates cos(C) once you know the angle; it does not take three sides and solve the triangle. The honest scope matters: if a tool promises triangle solving, check that it actually accepts side lengths.
Because spreadsheets and programming languages expect radians. =COS(60) computes the cosine of 60 radians, about -0.9524, not the cosine of 60 degrees. Convert first (=COS(RADIANS(60)) in Excel) or use this calculator with the unit set to degrees.
Add This Calculator to Your Website
References
- National Institute of Standards and Technology, Digital Library of Mathematical Functions, Section 4.15: trigonometric functions, including cosine graphics and shift identities. NIST DLMF: Trigonometric Functions, Section 4.15. Accessed October 2026.
- OpenStax, Precalculus: the other trigonometric functions and the unit-circle definitions behind cosine and its reciprocal relatives. OpenStax Precalculus: The Other Trigonometric Functions. Accessed October 2026.
- SpreadsheetCenter: the Excel COS function, which expects radians and returns the cosine used in spreadsheet work. SpreadsheetCenter: Excel COS Function. Accessed October 2026.
- ProRealAlgos Documentation: the COS function, its radian input convention, range of -1 to 1, and period of 2pi. ProRealAlgos Documentation: COS. Accessed October 2026.
- Union Test Prep, CLT Quantitative Reasoning study guide: unit circle coordinates, where the x-coordinate gives the cosine of the angle. Union Test Prep: Quantitative Reasoning Study Guide. Accessed October 2026.