cosine (idea)

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.