Circle/Trigonometric functions/Direct and problems/Introduction/Section

In ${}\mathbb {R} ^{2}$, the distance between two points ${}P,Q\in \mathbb {R} ^{2}$ is a positive real number (or equals ${}0$ in case the points coincide). If for both points the coordinates are given, say ${}P=(x_{1},y_{1})$ and ${}Q=(x_{2},y_{2})$, then the distance equals

${}d(P,Q)={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}\,.$

This equation rests on the Pythagorean theorems. In particular, the distance of every point ${}P=(x,y)$ to the zero point ${}(0,0)$ is

${\sqrt {x^{2}+y^{2}}}.$

As the coordinates are real numbers, so are the distances. If a point ${}M$ and a positive real number ${}r$ are given, then the set of all points in the plane, which have to ${}M$ the distance ${}r$, is the circle around ${}M$ with radius ${}r$. Written in coordinates, the definition is as follows.

Definition

Let ${}M=(a,b)\in \mathbb {R} ^{2}$ and ${}r\in \mathbb {R} _{+}$. Then the set

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is called the circle with center ${}M$ and radius ${}r$.

We stress that we mean the circumference and not the full disk. All circles are essentially the same, for the most important properties neither the center nor the radius are relevant. From this perspective, the unit circle is the simplest circle.

Definition

The set

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is called the unit circle.

The unit circle has radius ${}1$ and center ${}0=(0,0)$. In a naive approach, the trigonometric functions sine and cosine are defined with the help of the unit circle.

An "angle“ ${}\alpha$ at the zero point (measured starting with the positive "${}x$-axis“ and going "counterclockwise“) defines a ray. Since this ray has a unique intersection point ${}P(\alpha )=(x,y)$ with the unit circle, the angle defines a unique point on the unit circle. The coordinates of this point are by definition

${}P(\alpha )=(\cos \alpha ,\sin \alpha )\,,$

that is, the ${}x$-coordinate is given by cosine, and the ${}y$-coordinate is given by sine. Hence, many important properties are immediately clear:

1. We have

   ${}{\left(\cos \alpha \right)}^{2}+{\left(\sin \alpha \right)}^{2}=1\,.$

2. We have ${}\cos 0=1$ and ${}\sin 0=0$.
3. If the angle ${}\beta$ represents a quarter turn, then ${}\cos \beta =0$ and ${}\sin \beta =1$.
4. We have ${}\cos {\left(-\alpha \right)}=\cos \alpha$ and ${}\sin {\left(-\alpha \right)}=-\sin \alpha$. Here ${}-\alpha$ means the opposite angle and the opposite ray.
5. The values of sine and cosine repeat themselves after a complete turn.

The graphs of cosine and sine. The behavior is in principle clear from the naive definition. With the following analytic definition via series, we are able to compute the values exactly. In order to understand further properties like periodicity and period length ${}2\pi$, one has to study the analytic definition in more detail.

This definition of the trigonometric functions is intuitively clear, however, it is not satisfactory in several respects.

1. It is not clear how to measure an angle.
2. There is no analytic "computable“ expression how to calculate for a given angle the values of sine and cosine.
3. Hence, there is no fundament to prove properties about these functions.

Related with these deficits, is that we do not yet have a precise definition for the number ${}\pi$. This number equals the area of the unit circle and equals half of the length of the circumference. However, the concepts of an "area bounded by curves“ and of the "length of a curve“ are not easy. Hence, it is all in all better to define the trigonometric functions with the help of their power series, and then to prove step by step the relations with the circle. In this way, one can also introduce the number ${}\pi$ via these functions, and introduce the angle as the length of the circular arc, after we have established the length of a curve.