Question

a) Ein Punkt hat die kartesischen Koordinaten \(P=(3,3)\). Geben Sie die Polarkoordinaten an. b) Geben Sie die Gleichung für einen Kreis mit Radius \(R\) in Polarkoordinaten und kartesischen Koordinaten an. c) Geben Sie die Gleichung für die Spirale in Polarkoordinaten an.

Short answer

a) The polar coordinates are \(r = \sqrt{3^{2}+3^{2}}\) and \(\Theta = \text{atan2}(3,3)\). b) The equation of a circle in Cartesian coordinates is \( (x-h)^{2} + (y-k)^{2} = r^{2} \), and in Polar coordinates, it's \( r = R \). c) The equation of a spiral in Polar coordinates is \( r = a + b\Theta \).

Step-by-step solution

Convert Cartesian Coordinates to Polar Coordinates

To convert the Cartesian coordinates \(P=(3,3)\) into Polar coordinates, two formulas are used. The radial distance \(r\) is given by the formula \(r = \sqrt{x^{2}+y^{2}}\), and the angle \(\Theta\) (in radians) is given by \(\Theta = \text{atan2}(y,x)\), where \text{atan2} ensures the correct quadrant for \(\Theta\). From cartesian coordinates, \(x=3\) and \(y=3\). Applying these values into the formulas, gives the polar coordinates as \(r = \sqrt{3^{2}+3^{2}}\) and \(\Theta = \text{atan2}(3,3)\).

Equation of a Circle in Polar and Cartesian Coordinates

The general form of a circle's equation in Cartesian coordinates is \( (x-h)^{2} + (y-k)^{2} = r^{2} \), where \((h,k)\) are the coordinates of the circle's centre, and \(r\) is the radius. In Polar coordinates, the equation simplifies to \( r = R \), where \(R\) is the constant radius of the circle.

Equation of a Spiral in Polar Coordinates

The equation of a spiral in Polar coordinates can be represented as \( r = a + b\Theta \), where \(a\) and \(b\) are constants. The value of \(a\) defines the starting point of the spiral, and \(b\) represents the distance between successive turns in the spiral.

Key concepts

Cartesian Coordinates

Cartesian coordinates offer a simple way to represent points in an 2D plane. They are defined by two numbers \((x, y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance from the origin. These coordinates are easy to visualize in a grid system or a graph.

When locating a point on a plane, the Cartesian system is the go-to method for its straightforward representation.

• Horizontal axis is usually labeled as \(x\)-axis.
• Vertical axis is labeled as \(y\)-axis.

The point \((3,3)\) in Cartesian coordinates tells you that to reach this point from the origin, you go 3 units right and 3 units up.

Equation of a Circle

The equation of a circle in mathematics describes all the points that are a fixed distance, called the radius, away from a fixed point, called the center. In Cartesian coordinates, this equation can be written as: \[ (x-h)^{2} + (y-k)^{2} = r^{2} \]
Here, \((h, k)\) are the coordinates of the center. The radius \(r\) provides the circle's size.

In polar coordinates, describing a circle becomes even simpler:\[ r = R \]where \(R\) is the radius, assuming the center is at the origin. This highlights the circle as a set of points at distance \(R\) from the origin, no matter the angle \(\Theta\).

• Polar coordinates can often simplify equations for symmetric shapes centered at the origin.
• Understanding both representations is useful for different applications, such as geometry and physics.

Spiral Equation

Spirals are fascinating geometric shapes that are often seen in nature and technology. In polar coordinates, a spiral is represented with the equation:\[ r = a + b\Theta \]where \(a\) and \(b\) are constants.

Here’s what these constants mean:

• \(a\): Determines where the spiral starts. When \(\Theta = 0\), \(r = a\) means this is your initial point on the spiral.
• \(b\): Controls the distance between the loops of the spiral. A larger \(b\) means wider turns.

As \(\Theta\) increases, the spiral grows larger, tracing outward in the plane. Spirals in polar coordinates are straightforward to describe due to the natural way angles and distances are involved, making them perfect for circular and rotational patterns.

Coordinate Conversion

Converting coordinates from Cartesian to Polar and vice versa is a crucial skill in many mathematics and physics problems. This conversion is key to solving problems that involve circular motion or require different perspectives on data.

The conversion involves:

• Calculating the radial distance \(r\) using the formula: \[ r = \sqrt{x^2+y^2} \]
• Finding the angle \(\Theta\) with: \[ \Theta = \text{atan2}(y, x) \]

For example, converting point \((3,3)\) from Cartesian to Polar coordinates involves finding the distance \(r\) and angle \(\Theta\).

• This results in \(r = \sqrt{3^2 + 3^2} = \sqrt{18}\).
• \(\Theta = \text{atan2}(3,3)\) which adequately locates the point within the appropriate quadrant.

Understanding this process equips you with the flexibility to transition between different coordinate systems, making analyses easier to perform in various contexts.