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The Cosine function (cos(x))

The cosine function, often denoted as cos(x), is one of the fundamental functions in trigonometry. The function relates the angle of a right triangle to the ratio of the length of the adjacent side over the length of the hypotenuse.

Here are some key points about the cosine function:

  • Definition on the Unit Circle: For an angle x, measured in radians, cos(x) represents the x-coordinate of a point on the unit circle (a circle with a radius of 1 unit). The unit circle is centered at the origin (0,0) in the coordinate plane.
  • Periodicity: The cosine function is periodic with a period of 2π radians, which means that cos(x) = cos(x + 2πk) for any integer k. This periodic nature also means that the cosine function repeats its values every 360 degrees (since 2π radians equals 360 degrees).
  • Range: The range of the cosine function is from -1 to 1. That is, cos(x) will always yield a value within this interval.
  • Even Function: The cosine function is an even function, which means it is symmetric with respect to the y-axis. This implies that cos(-x) = cos(x) for all x.
  • Key Angles: The values of cos(x) for key angles are often memorized. For example, cos(0) = 1, cos(π/2) = 0, cos(π) = -1, cos(3π/2) = 0, and cos(2π) = 1.
  • Relationship to Sine: The cosine function is related to the sine function (sin(x)) through the identity cos(x) = sin(π/2 - x). They are also related by the Pythagorean identity sin^2(x) + cos^2(x) = 1.
  • Graph: The graph of cos(x) is a wave-like curve that starts at (0,1), moves down to (π/2,0), to (π,-1), up to (3π/2,0), and back to (2π,1), and then repeats.
  • Applications: The cosine function is used in various fields such as physics, engineering, and computer science. It describes oscillations, waves, and it can be used to model periodic phenomena.

In calculus, the derivative of cos(x) with respect to x is -sin(x), and the integral of cos(x) with respect to x is sin(x) plus a constant of integration.

What is Degrees (deg °) and Radians (rad) ?

In the context of trigonometric functions and other mathematical applications, "Degrees" and "Radians" refer to two different units for measuring angles:

  • "Degrees" are a measure of angle using the familiar system where a complete circle is divided into 360 degrees.
  • "Radians" are a measure of angle used in mathematics, especially in trigonometry and calculus, where a complete circle corresponds to 2π radians. One radian is defined as the angle created by an arc that is equal in length to the radius of the circle.

To convert between degrees and radians, the following two formulas can be used:

  • From degrees to radians:
    radians = degrees ×
  • From radians to degrees:
    degrees = radians ×

Table of common cosine values

Angle (°)Angle (Radians)cos(angle)cos(angle)

See also
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