# Equivalence of Definitions of Sine and Cosine

## Theorem

In the following, $\theta$ understood to take values in $\hointr 0 {2 \pi}$.

### The following definitions of the concept of **Sine Function** are equivalent:

#### 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 $x$-axis.

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

Hence in the first quadrant, the **sine** is positive.

#### Analytic Definition

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

\(\ds \forall x \in \R: \, \) | \(\ds \sin x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\) |

### The following definitions of the concept of **Cosine Function** are equivalent:

#### 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.

#### Analytic Definition

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

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

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

## Proof

Consider the following vector-valued function $\mathbf f : \R \to {\closedint {-1} 1}^2$:

- $\map {\mathbf f} t = \tuple {\cos t, \sin t}$

where $\cos t$ and $\sin t$ are defined analytically.

Then, for any $t$ the distance to the origin is:

\(\ds \norm {\map {\mathbf f} t - \bszero}\) | \(=\) | \(\ds \norm {\map {\mathbf f} t}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sqrt {\cos^2 t + \sin^2 t}\) | Definition of Euclidean Norm | |||||||||||

\(\ds \) | \(=\) | \(\ds 1\) | Sum of Squares of Sine and Cosine |

Therefore, $\map {\mathbf f} t$ always lies on the unit circle.

For arbitrary $\theta \in \hointr 0 {2 \pi}$, the arc length on $\closedint 0 \theta$ is:

\(\ds s\) | \(=\) | \(\ds \int_0^\theta \sqrt {\paren {\frac \d {\d t} \cos t}^2 + \paren {\frac \d {\d t} \sin t}^2} \rd t\) | Arc Length for Parametric Equations | |||||||||||

\(\ds \) | \(=\) | \(\ds \int_0^\theta \sqrt {\paren {-\sin t}^2 + \paren {\cos t}^2} \rd t\) | Derivative of Cosine Function and Derivative of Sine Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \int_0^\theta \rd t\) | Sum of Squares of Sine and Cosine | |||||||||||

\(\ds \) | \(=\) | \(\ds \theta\) | Definite Integral of Constant |

Thus, by the definition of radians, the angle made by $\map {\mathbf f} \theta$ with the $x$-axis is $\theta$.

A straight line segment from $\tuple {0, \sin \theta}$ to $\tuple {\cos \theta, \sin \theta}$ is perpendicular to the $y$-axis.

Its length is $\size {\cos \theta}$, and it is on the side of the $y$-axis corresponding to the sign of $\cos \theta$.

But that is the unit circle definition for cosine.

$\Box$

Similarly, a straight line segment from $\tuple {\cos \theta, 0}$ to $\tuple {\cos \theta, \sin \theta}$ is perpendicular to the $x$-axis, with length $\size {\sin \theta}$ and appropriate direction.

Again, that is the unit circle definition for sine.

$\blacksquare$