The cosine function, written as cos theta, is a fundamental trigonometric function. In a right triangle, cosine of an angle equals the length of the adjacent side divided by the length of the hypotenuse. This ratio remains constant for any given angle, regardless of the triangle's size.
The mathematical definition of cosine is the ratio of the adjacent side to the hypotenuse. For example, in a triangle with adjacent side of length 3 and hypotenuse of length 5, cosine theta equals 3 divided by 5, which is 0.6. This ratio remains constant for this specific angle in any similar triangle.
On the unit circle, cosine has a geometric interpretation. For any angle theta, cosine equals the x-coordinate of the corresponding point on the circle. As we rotate around the circle, the cosine value changes smoothly from 1 to negative 1 and back to 1, creating the familiar wave pattern.
The graph of the cosine function shows its periodic nature. Starting at 1 when x equals 0, it decreases to 0 at π/2, reaches -1 at π, returns to 0 at 3π/2, and completes the cycle back at 1 when x equals 2π. This pattern repeats indefinitely, making cosine a periodic function with period 2π.
The cosine function has several important properties. It's an even function, meaning it's symmetric about the y-axis. Its range is from negative 1 to positive 1, and it has a period of 2π. Cosine is widely used in physics for describing waves and oscillations, in engineering for signal processing, and in computer graphics for rotations and transformations.