Zeichnen Sie die Ortskurven der folgenden komplexen Zeiger: a) \(z(t)=a \cdot \cos t+j \cdot b \cdot \sin t \quad(0 \leqslant t<2 \pi)\) b) \(\quad z(t)=2 \cdot \cos ^{2} t+j \cdot \sin (2 t) \quad\left(0 \leqslant t<\frac{\pi}{2}\right)\)

An ellipse is a geometric figure defined as the locus of all points such that the sum of the distances from any point on the ellipse to two fixed points (called the foci) is constant. In simpler terms, it's an elongated circle.
In the given problem, the ellipse is defined using the parametric equations:
\[ x = a \cos t \text{ and } y = b \sin t \]
This means as variable \(t\) moves from \(0\) to \(2\pi\), \(x\) and \(y\) trace out an ellipse.
The general form of an ellipse equation in Cartesian coordinates is:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\]
Here, \(a\) and \(b\) are the semi-major and semi-minor axes, giving the ellipse its shape. This shape appears when a plane intersects a cone at an angle not parallel to the base.

Parametric equations express the coordinates of the points of a curve as functions of a variable. For instance, in exercise (a), we have:
\[ x = a \cos t \text{ and } y = b \sin t.\] These equations allow us to describe the position of a point on the curve with a single parameter \(t\).

For the second exercise (b), the parametric equations are given as:
\[ x = 2 \cos^2 t \text{ and } y = \sin (2t).\]
By using parametric equations, we can easily generate points to be plotted on a graph, helping visualize and understand the curve's shape. They are especially useful for describing complex curves and are extensively used in computer graphics and animations.

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. They simplify the process of solving trigonometric equations.
For example, in step (5) of the solution, we used the identity:

\[ \cos^2 t = \frac{1 + \cos (2t)}{2} \]
This identity allows us to express \(2 \cos^2 t\) in terms of \(\cos(2t)\). This simplification helps understand how the components of the parametric equations relate and describe the curve. Knowing and applying these identities is crucial for solving many problems in trigonometry and calculus.

The imaginary unit \(j\) is defined as the square root of -1. It is represented as:
\[ j^2 = -1.\]
Complex numbers are written in the form \(z = x + jy\), where \(x\) is the real part, and \(jy\) is the imaginary part.
In our cases:

• For exercise (a), \( x = a \cos t \) and \( y = b \sin t \).
• For exercise (b), \( x = 2 \cos^2 t \) and \( y = \sin (2t) \).

By representing complex numbers in this way, we can plot them on the Cartesian plane and analyze their behavior graphically. The imaginary unit is a foundational concept in complex number theory and is used extensively in engineering and physics.

The Cartesian plane is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Each point on the plane is identified by an ordered pair \((x, y)\).

• In exercise (a), the points are \((a \cos t, b \sin t)\).
• In exercise (b), the points are \((2 \cos^2 t, \sin (2t))\).

Plotting these points provides a visual representation of the curves defined by the parametric equations.
The Cartesian plane is a critical tool in graphing functions and analyzing geometrical shapes. Understanding how to plot and interpret curves on this plane is essential for solving many mathematical problems.