Calculator Cos 45 in Degrees

What is cos 45 degrees?

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse. For 45 degrees, this ratio is a well-known mathematical constant.

In trigonometry, cos(45°) is exactly √2/2, which is approximately 0.7071. This value is derived from the properties of a 45-45-90 right triangle, where the two legs are of equal length and the hypotenuse is √2 times the length of each leg.

For a 45-45-90 triangle with legs of length 1, the hypotenuse is √2. Therefore:

How to calculate cos 45 degrees

Calculating cos 45 degrees involves understanding the relationship between the sides of a right-angled triangle. Here's a step-by-step method:

1. Draw a right-angled triangle with both non-right angles equal to 45 degrees.
2. Label the two legs adjacent to the 45° angles as length 1.
3. Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(1² + 1²) = √2.
4. Apply the cosine formula: cos(45°) = adjacent side / hypotenuse = 1 / √2.
5. Rationalize the denominator: 1 / √2 = √2/2.

This method demonstrates why cos(45°) is exactly √2/2. The calculator on this page performs this calculation automatically for any angle you input.

Cos 45 degrees example

Let's use cos 45 degrees in a practical scenario. Suppose you're designing a ramp that needs to rise 1 meter for every 1 meter of horizontal distance. The angle of elevation θ can be found using the arctangent function:

Now, if you want to find the horizontal distance (adjacent side) when the vertical rise (opposite side) is 2 meters, you can use the tangent function:

Since tan(45°) = 1, the horizontal distance x is 2 meters. This demonstrates how cos 45 degrees relates to other trigonometric functions in practical applications.