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:

`c`^{2} = `a`^{2} + `b`^{2} − 2`ab` 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 2`θ` = cos^{2} `θ` − sin^{2} `θ` = 2 cos^{2} `θ` − 1

cos^{2} `θ` + sin^{2} `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 cos^{2} `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.