# Definition:Cosine

## Contents

## Definition

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **cosine** of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.

### Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.

Let $P = \left({x, y}\right)$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $y$-axis.

Then the **cosine** of $\theta$ is defined as the length of $AP$.

### Real Numbers

The real function $\cos: \R \to \R$ is defined as:

\(\displaystyle \cos x\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots\) | $\quad$ | $\quad$ |

### Complex Numbers

The complex function $\cos: \C \to \C$ is defined as:

\(\displaystyle \cos z\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\) | $\quad$ | $\quad$ |

## Linguistic Note

As with **sine**, the word **cosine** derives (erroneously, via a mistranslation) from the Latin **sinus** which (among other things) means **fold**, **curve**, **winding** or **bay**.

The **co-** prefix, as with similar trigonometric functions, is a reference to complementary angle: see Cosine of Complement equals Sine.

It is pronounced with the emphasis on the first syllable: ** co-sign**.

$\cos x$ is voiced **cosine (of) $x$**, or (as written) **cos $x$** (pronounced either **coss** or **coz** depending on preference).

## Historical Note

The the symbol $\cos$ for cosine appears to have been invented by William Oughtred in his $1657$ work *Trigonometrie*, although some authors attribute it to Euler.

## Also see

- Definition:Sine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Secant
- Definition:Cosecant

- Results about
**the cosine function**can be found here.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles