Category:Examples of Cosine Function

This category contains examples of Cosine Function.

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 plane.

Let $P = \tuple {x, y}$ 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 $AP$.

Hence in the first quadrant, the cosine is positive.

Real Numbers

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

\(\ds \cos x\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }\)

\(\ds \)

\(=\)

\(\ds 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots\)

Complex Numbers

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

\(\ds \cos z\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\)

\(\ds \)

\(=\)

\(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\)