Cosine Function

Cosine Function

Function

cos(θ)

Summary

Given the angle of a right triangle as input, returns the ratio of the triangle's adjacent side over its hypotenuse.

Input

Name Description
θ The symbol θ (theta) is a number representing the angle of a right triangle.

Output

Returns the ratio of the right triangle's adjacent side over its hypotenuse.

Examples

Cosine of π/6

Special Right Triangle 30 60 90

The cosine of π/6 radians or 30° is equal to the square root of 3 divided by 2. This is calculated from the definition of cosine: the adjacent side over the hypotenuse side of the right triangle.

Cosine of π/4

Special Right Triangle 45 45 90

The cosine of π/4 radians or 45° is equal to 1 divided by the square root of 2. This is calculated from the definition of cosine: the adjacent side over the hypotenuse side of the right triangle.

Cosine of π/3

Special Right Triangle 60 30 90

The cosine of π/3 radians or 60° is equal to 1 divided by 2. This is calculated from the definition of cosine: the adjacent side over the hypotenuse side of the right triangle.

Cosine of 3 4 5 Triangle

Right Triangle 3 4 5

Explanation

The graph of the cosine function is shown below. The x-axis represents the input of angles from 0 to τ (tau) radians. The y-axis represents the ratio of the right-triangle’s adjacent side over its hypotenuse. Note: Radians are a unit of measure used to measure angles - a full rotation (360°) in radians is equal to approximately 6.28 radians or τ radians.

Graph of Cosine Function

Geometrically, the function can be visualized with a right-triangle of hypotenuse length 1 which forms the unit circle. The shape of this right triangle is a similar triangle to all triangles corresponding to the input angle formed between the x-axis and hypotenuse of the triangle. Click and drag either of the two control points to see how the input angle changes the output of the function.

Since the right triangle in the unit circle is the general case of the cosine function, the definition simplifies to correspod directly with the length of the adjacent side of the triangle. In applications, the result of the cosine function is often scaled by the hypotenuse of the right triangle.

Special Right Triangle 60 30 90 Unit Circle Cosine at PI Thirds

Good Links

Angle | concept
Radians | concept
Right Triangle | concept