Definition:Cosine

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Definition

Definition from Triangle

SineCosine.png

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.


CosineFirstQuadrant.png


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

  • Results about the cosine function can be found here.


Sources