Introduction
This cosine calculator finds the exact value of cos(α) for any angle you type in. Enter your angle in degrees or radians, and the tool gives you the answer right away. It also shows a step-by-step solution so you can see how the math works.
The cosine function is one of the main functions in trigonometry. It tells you the ratio of the side next to an angle over the longest side (the hypotenuse) in a right triangle. You use it in geometry, physics, engineering, and many other fields. The value of cosine always falls between −1 and 1.
You can pick how many decimal places you want in your result, copy the answer with one click, and view a graph that plots the cos wave around your chosen angle. There is also an advanced expression evaluator that lets you type full math expressions like cos(60deg) + cos(90deg) and get a combined result.
How to Use Our Cosine Calculator
Enter an angle and choose your settings below. The calculator will return the exact cosine value (cos) of that angle along with a step-by-step solution and a graph.
Angle (α): Type the angle you want to find the cosine of into the input box. You can use whole numbers or decimals. Click the π button to insert the value of pi.
Angle Unit: Pick Degrees (°) or Radians (rad) from the dropdown menu to match your angle. If you need help converting between units, remember that a full circle is 360° or 2π radians.
Decimal Places: Choose how many decimal places you want in your answer. The default is 6.
Calculate: Press the Calculate button or hit Enter to get your result. The cosine value, a full formula, step-by-step work, and a graph will appear right away.
Copy: Click the Copy button next to the result to copy the cosine value to your clipboard.
Advanced Expression: Use the expression box to solve more complex problems like cos(45deg) + 1. You can combine cos() with addition, subtraction, multiplication, division, and exponents. Press Calculate or hit Enter to evaluate.
What Is Cosine?
Cosine (often written as cos) is one of the main functions in trigonometry. It connects an angle of a right triangle to the ratio of two of its sides. Specifically, the cosine of an angle equals the length of the adjacent side (the side next to the angle) divided by the length of the hypotenuse (the longest side of the triangle).
The Cosine Formula
The basic formula is:
cos(α) = adjacent ÷ hypotenuse
Here, α (alpha) is the angle you are working with. The result of cos(α) is always a number between −1 and 1.
Degrees vs. Radians
Angles can be measured in two common units: degrees and radians. A full circle is 360 degrees or 2π radians. To convert degrees to radians, multiply the degree value by π/180. This calculator handles both units for you automatically.
Common Cosine Values
- cos(0°) = 1
- cos(30°) = √3/2 ≈ 0.866025
- cos(45°) = √2/2 ≈ 0.707107
- cos(60°) = 0.5
- cos(90°) = 0
- cos(180°) = −1
- cos(360°) = 1
Where Is Cosine Used?
Cosine shows up in many real-world tasks. Engineers use it to calculate forces and angles in structures. Physicists use it to study waves, sound, and light. Game developers use it to move objects on a screen. Pilots and sailors use it for navigation. Any time you need to find a missing side or angle in a triangle, cosine can help. For triangles that are not right-angled, the Law of Cosines and the Law of Sines solve for unknown sides and angles. Cosine is also essential in projectile motion calculations, where it breaks a launch velocity into its horizontal component.
The Cosine Wave
When you plot cosine values on a graph, you get a smooth, repeating wave. This wave starts at 1 when the angle is 0°, drops to 0 at 90°, reaches −1 at 180°, returns to 0 at 270°, and comes back to 1 at 360°. This pattern repeats forever in both directions. The shape is called a cosine wave, and it is the same as a sine wave shifted by 90°. Cosine waves are closely tied to concepts like frequency, which describes how many cycles a wave completes per second.