To solve the problem, use the dot product for two vectors: $vec a =(cos theta, sin theta)^T$ and $vec b =(cos phi, sin phi)^T$. This will give the equation $vec a cdot vec b= |a||b| cos (theta-phi)=cos (theta-phi)=cos theta cos phi +sin theta sin phi$. Then, find $sin(theta -phi)$ using the equation $sin(theta -phi)=sqrt{1-cos^2 (theta-phi)}$.

The question is to solve a system of equations symbolically for two variables, specifically for $x$ and $y$, using sympy. In radians, the arc length in a unit circle is equal to the angle $x$. For small angles, we have $sin x approx x$. This can be used to simplify calculations.

Question:

As I delve into the Multivariable course, I am currently expanding my knowledge on spherical coordinates. However, I am faced with a challenge when it comes to dealing with

Learn how to graph

shapes.

This is the problem:

What shapes are described when…?

Solution:

A sphere of radius 1 is represented by $rho = 1$.

The cone has an angle of $frac{pi}{3}$ and is represented by $phi = frac{pi}{3}$.

For $theta = frac{pi}{4}$, the cross-section with a z-axis diameter is shown in

Semi-circular

.

d) $rho = cos{(phi)}$ : ?

e) $rho = cos{(2theta)}$ : …?

Do the last two -d) and e) require a verbal description and if so, what would it be?

Solution:

In the case where $phi$ is a cone having an angle of $pi/3$, the following holds true:

d) $rho=cosphi$:

By multiplying both terms with $rho$, we obtain $rho^2=rhocosphi$, which can be simplified to $x^2+y^2+z^2=z$. This equation represents a sphere with a radius of $1/2$ and a center at $(0,0,1/2)$. Please refer to the image below for a visual representation:

e) $rho=cos2theta$:

The

cartesian equation

may be located here eventually, though it won’t be useful since it’s not a typical surface.

All of the options, a), b), and c), are accurate. To verify, locate the

cartesian equations

.