# 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.

$\blacksquare$