Chapter 15: Problem 3

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

Expert verified

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.