Cos Calculator in Degrees

The cosine of an angle is a fundamental trigonometric function that relates the angle to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This calculator computes the cosine of an angle given in degrees, providing both the numerical result and a visual representation of the cosine function.

What is Cosine in Degrees?

Cosine is one of the primary trigonometric functions, along with sine and tangent. In a right-angled triangle, the cosine of an angle θ (theta) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The cosine function extends beyond right-angled triangles to all angles on the unit circle.

When working with degrees, the cosine function is periodic with a period of 360°, meaning cos(θ) = cos(θ + 360°n) for any integer n. The cosine of 0° is 1, and the cosine of 90° is 0. The function is even, meaning cos(-θ) = cos(θ).

Cosine values range between -1 and 1 for all real numbers. In degrees, cosine is positive in the first and fourth quadrants (0° to 90° and 270° to 360°), negative in the second and third quadrants (90° to 270°), and zero at 90° and 270°.

How to Use the Cos Calculator

Using the cosine calculator is straightforward. Simply enter the angle in degrees that you want to find the cosine of, then click the "Calculate" button. The calculator will display the cosine value and a chart showing the cosine function for angles around your input.

The calculator accepts angles from -360° to 360°, but the cosine function is periodic, so results will repeat every 360°.

Cosine Formula

The cosine of an angle θ in degrees can be calculated using the following formula:

cos(θ) = adjacent / hypotenuse

For angles beyond the first quadrant (0° to 90°), the sign of the cosine depends on the quadrant:

First quadrant (0° to 90°): cos(θ) is positive
Second quadrant (90° to 180°): cos(θ) is negative
Third quadrant (180° to 270°): cos(θ) is negative
Fourth quadrant (270° to 360°): cos(θ) is positive

Worked Examples

Example 1: Calculating cos(30°)

For a 30° angle in a right-angled triangle with sides 1, √3, and 2:

Adjacent side = √3
Hypotenuse = 2

cos(30°) = √3 / 2 ≈ 0.8660

Example 2: Calculating cos(180°)

For a 180° angle, the adjacent side is -1 and the hypotenuse is 1:

cos(180°) = -1 / 1 = -1

Example 3: Calculating cos(270°)

For a 270° angle, the adjacent side is 0 and the hypotenuse is 1:

cos(270°) = 0 / 1 = 0

FAQ

What is the range of cosine values?

The cosine of any angle in degrees ranges from -1 to 1. The maximum value of 1 occurs at 0°, 360°, and -360°, while the minimum value of -1 occurs at 180°.

How does cosine differ from sine?

Cosine and sine are complementary functions. While cosine relates the adjacent side to the hypotenuse, sine relates the opposite side to the hypotenuse. They are also phase-shifted by 90°: sin(θ) = cos(θ - 90°).

Can cosine be negative?

Yes, cosine is negative in the second and third quadrants (90° to 270°). In these quadrants, the adjacent side is negative relative to the standard orientation.

What is the cosine of 0°?

The cosine of 0° is 1, as it corresponds to the ratio of the adjacent side (1) to the hypotenuse (1) in the standard position.