91Ó°ÊÓ

Spherical coordinates are a 3-dimensional coordinate system where a point in space is represented by three coordinates (\(\rho, \varphi, \theta\)). \(\rho\) is the distance from the origin, \(\varphi\) is the polar angle measured from the positive z-axis, and \(\theta\) is the azimuthal angle measured from the positive x-axis in the xy-plane. These coordinates can be related to Cartesian coordinates \((x, y, z)\) using the following equations: $$ x = \rho \sin\varphi \cos\theta \\ y = \rho \sin\varphi \sin\theta \\ z = \rho \cos\varphi $$ Step 2: Analyze Given Set in Spherical Coordinates

The given set has two conditions: 1. \(\rho = 2 \csc \varphi\) 2. \(0 < \varphi < \pi\) The first constraint represents a relationship between \(\rho\) and \(\varphi\), while the second one is an interval for \(\varphi\). Step 3: Rewrite \(\rho\) in terms of \(\varphi\)

We know that \(\csc \varphi = \frac{1}{\sin \varphi}\), so we can rewrite the given constraint for \(\rho\) as follows: $$ \rho = 2 \csc \varphi = 2\frac{1}{\sin \varphi} $$ Step 4: Sketch the Given Set

To sketch the given set, we'll focus on the constraint \(\rho = 2\frac{1}{\sin \varphi}\) and the range \(0 < \varphi < \pi\). We can notice that, when approaching the poles where \(\varphi \to 0\) or \(\varphi \to \pi\), the value of \(\rho\) will tend to infinity. In other words, we have some sort of annular surface extending outwards from the origin. Since there's no constraint on the azimuthal angle \(\theta\), it means that this set will rotate around the z-axis. Therefore, the given set represents a cone-like structure extending outwards from the positive z-axis to the negative z-axis. The angle between this cone-like structure and xy-plane will remain constant. And there you have it! The given set in spherical coordinates represents a double cone-like structure open at both ends, with a constant angle between the structure and the xy-plane.