find an equation in cylindrical coordinates for the rectangular equation y = x^2. entry tip #1: to enter θ…

find an equation in cylindrical coordinates for the rectangular equation y = x^2. entry tip #1: to enter θ, type \theta\. example: r*sin(theta) = r*sin(θ) entry tip #2: to enter a squared trig function, wrap the outside of the trig function with parentheses. example: (sin(theta))^2 = sin^2(θ) question help: message instructor submit question

find an equation in cylindrical coordinates for the rectangular equation y = x^2. entry tip #1: to enter θ, type \theta\. example: r*sin(theta) = r*sin(θ) entry tip #2: to enter a squared trig function, wrap the outside of the trig function with parentheses. example: (sin(theta))^2 = sin^2(θ) question help: message instructor submit question

Answer

Explanation:

Step1: Recall cylindrical - rectangular conversion

In cylindrical coordinates, $x = r\cos\theta$, $y = r\sin\theta$.

Step2: Substitute into the given equation

Substitute $x = r\cos\theta$ and $y = r\sin\theta$ into $y=x^{2}$. We get $r\sin\theta=(r\cos\theta)^{2}$.

Step3: Simplify the equation

Expand $(r\cos\theta)^{2}=r^{2}\cos^{2}\theta$. So the equation becomes $r\sin\theta = r^{2}\cos^{2}\theta$. If $r\neq0$, we can divide both sides by $r$ to obtain $\sin\theta=r\cos^{2}\theta$.

Answer:

$\sin\theta = r\cos^{2}\theta$