One of the fundamental trigonometric functions. The derivative of sin, a.k.a. sine. Commonly abbrieviated cos. An operative function in many programming languages. Notations vary, i.e., cos-1x may mean 1/cosx or may mean arccosx. Useful in all branches of mathematics. Hyperbolic equivalent in cosh. Also equivalent to:


This means that cos(x)=cosh(ix).

One possible series for finding cos(x) is

cos(x) |-> 1 - x2/2! + x4/4! - x6/6! + ...

The first term, of 1, is simply x0/0!, so it does fit the pattern of the other terms; this helps to remember the series. It is obvious from this that cos(0) = 1, since all the other terms evaluate to 0.

Furthur, it is apparent from inspection that imaginary values of x will always produce real answers, since x is only raised to even powers. This is the reverse of sine, where imaginary x will always produce imaginary output.

Note that this definition of cosine requires an input in radians, not degrees, with which some are more familiar. To convert to degrees, simply multiply by 180/pi.

Compare this series to the series for hyperbolic cosine.

Here are some more useful facts about cosine, some gathered together from other nodes, others apparently not yet noded. The cosine of an angle (measured anticlockwise from the x-axis) is the proportion it projects onto the x-axis. A line of length 1 in a direction θ has a horizontal component of cos θ and a vertical one of sin θ.

In a right-angled triangle the cosine of one of the acute angles is the ratio of the adjacent side to the hypotenuse, which sounds better in pictures:

                     / |
                    /  |
                   /   |
                  /    |
     hypotenuse  /     |  opposite
                /      |
               /       |
              /       _|
        angle   adjacent
The graph of cosine is the same sinusoidal curve as for sine, but translated sideways, with the peaks coming π/2 earlier.
The derivative of cos is −sin, and the general antiderivative is sin.

The law of cosines: In any triangle, the length of a side is related to the opposite angle and the other two lengths by the following:

c2 = a2 + b2 − 2ab cos C
where we are writing C for the angle opposite side c. When C is a right angle, c is the hypotenuse and this reduces to the Pythagorean theorem.

Exact values of sine, cosine, and tan: There is an easy-to-remember progression of exact values for the three most important acute angles, 30°, 45°, and 60°:

cos 0° = √4 / 2 = 1
cos 30° = √3 / 2 ≈ 0.8660
cos 45° = √2 / 2 = 1/2 ≈ 0.7071
cos 60° = √1 / 2 = 0.5
cos 90° = √0 / 2 = 1
Note this is not simply formulaic: 15° and 75° don't fit in so neatly, but they're less often used. Sines work in the same way, but upwards from 0 to 4. The other important value to know is cos 180° = cos π = −1.

Other identities:

cos (−θ) = cos θ
cos (θ + φ) = cos θ cos φ − sin θ sin φ
cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1
cos2 θ + sin2 θ = 1
cos (θ + 2π) = cos θ
cos θ = sin (&pi/2 − θ)
cos (π − θ) = −cos θ

secant: The reciprocal of cosine is called secant, abbreviated sec. This isn't greatly important as a function in its own right, except as a notational convenience: although the square of cos x is written cos2 x, its reciprocal is never written as cos−1 x, that notation being reserved for its inverse function.

arccosine: The inverse of the sine function is arccosine, symbol cos−1 or arcos or arccos. Since cos is periodic, its inverse is not uniquely defined as a function. Restricting cos to the interval [0, π] makes it a one-to-one mapping onto the interval [−1, 1], so we can define a principal arccosine function, symbolized Arccos or Cos−1. So Arccos 1/√2 = π/4.

Co"sine (k?"s?n), n. [For co. sinus, an abbrev. of L. complementi sinus.] Trig.

The sine of the complement of an arc or angle. See Illust. of Functions.


© Webster 1913.

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