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Convert the equation to polar form. (Use variables $r$ and $\theta$ as needed.) $\newline$ $y=6x^{2}$

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Convert complex numbers between rectangular and polar form

Full solution

Q. Convert the equation to polar form. (Use variables

r

and

\theta

as needed.)

\newline

y=6x^{2}

Recalling Coordinate Relationships: We start by recalling the relationships between rectangular coordinates $(x, y)$ and polar coordinates $(r, \theta)$ : $x = r \cos \theta$ and $y = r \sin \theta$ . We will use these to convert the given rectangular equation $y = 6x^2$ into polar form.
Substitute into Equation: Substitute $x$ with $r \cos \theta$ and $y$ with $r \sin \theta$ into the equation $y = 6x^2$ to get $r \sin \theta = 6(r \cos \theta)^2$ .
Simplify the Equation: Simplify the equation by squaring $r \cos \theta$ to get $r \sin \theta = 6r^2 \cos^2 \theta$ .
Eliminate $r$ Term: Divide both sides of the equation by $r$ to eliminate the $r$ term on the left side, being careful to avoid division by zero. This gives us $\sin \theta = \frac{6r \cos^2 \theta}{r}$ .
Use Pythagorean Identity: Recognize that $\cos^2 \theta$ can be written as $(1 - \sin^2 \theta)$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ . Substitute this into the equation to get $\sin \theta = 6r(1 - \sin^2 \theta)$ .
Distribute $6r$ : Distribute the $6r$ on the right side to get $\sin \theta = 6r - 6r \sin^2 \theta$ .
Rearrange Equation: Rearrange the equation to isolate terms involving $r$ on one side. This gives us $6r \sin^2 \theta + \sin \theta - 6r = 0$ .
Correcting Mistake: Notice that we have a quadratic equation in terms of $r \sin \theta$ . Let's set $r \sin \theta = u$ , then our equation becomes $6u^2 + u - 6r = 0$ .
Correcting Mistake: Notice that we have a quadratic equation in terms of $r \sin \theta$ . Let's set $r \sin \theta = u$ , then our equation becomes $6u^2 + u - 6r = 0$ . Now, we need to solve this quadratic equation for $u$ . However, we realize that we have made a mistake in the previous step. The term $\sin \theta$ is not multiplied by $r$ , so we cannot simply substitute $r \sin \theta = u$ . We need to correct this and go back to the step before the substitution.

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