Q. 4 - Calculus

Question

Consider the point in the plane given by polar coordinates $(r,\theta )=\left(2,\frac{\pi }{3}\right)$.

(a) Express this point in polar coordinates where $r=2$, but $\theta \ne \frac{\pi }{3}$, if possible.

(b) Express this point in polar coordinates where $\theta =\frac{\pi }{3}$, but $r\ne 2$, if possible.

(c) Express this point in polar coordinates where $r\ne 2$ and $\theta \ne \frac{\pi }{3}$, if possible.

Step-by-Step Solution

Answer

(a). The answer is $\left(2,\frac{7\pi }{3}\right)$.

(b). The answer is $\left.\left(-2,\frac{\pi }{3}\right)\right)$.

(c). The answer is $\left(-2,\frac{7\pi }{3}\right)$.

Part(a) Step 1: Given information

The point $\left(2,\frac{\pi }{3}\right)$ where $r\ne 2$ but $\theta =\frac{\pi }{3}$.

Part(a) Step 2: calculation

Consider the polar $(r,\theta )=\left(2,\frac{\pi }{3}\right)$ coordinate.

Every point in polar coordinates will have infinitely many representations in polar plane.

The point $\left(2,\frac{\pi }{3}\right)$ can be written as ,

$$\begin{gathered} \left(2,\frac{\pi }{3}\right)=\left(2,\frac{\pi }{3}+2\pi \right) \\ =\left(2,\frac{7\pi }{3}\right) \end{gathered}$$

Thus, $\left(2,\frac{7\pi }{3}\right)$ is the required point where $r=2$ but $\theta \ne \frac{\pi }{3}$.

Therefore, the answer is $\left(2,\frac{7\pi }{3}\right)$.

Part(b) Step 1: Given information

The point $\left(2,\frac{\pi }{3}\right)$ where $r\ne 2$ but $\theta =\frac{\pi }{3}$.

Part(b) Step 2: calculation

Consider the polar $(r,\theta )=\left(2,\frac{\pi }{3}\right)$ coordinate.

Every point in polar coordinates will have infinitely many representations in polar plane.

The point $\left(2,\frac{\pi }{3}\right)$ can be written as $\left(-2,\frac{\pi }{3}\right)$,

Thus, $\left(-2,\frac{\pi }{3}\right)$ is the required point where $r\ne 2$ but $\theta =\frac{\pi }{3}$.

Therefore, the answer is $\left.\left(-2,\frac{\pi }{3}\right)\right)$.

Part(c) Step 1: Given information

The point $\left(2,\frac{\pi }{3}\right)$ where $r\ne 2$ but $\theta \ne \frac{\pi }{3}$.

Part(c) Step 2: Calculation

Consider the polar$(r,\theta )=\left(2,\frac{\pi }{3}\right)$ coordinate.

Every point in polar coordinates will have infinitely many representations in the polar plane.

The point $\left(2,\frac{\pi }{3}\right)$ can be written as,

$$\begin{gathered} \left(-2,\frac{\pi }{3}+2\pi \right) \\ \left(-2,\frac{7\pi }{3}\right) \end{gathered}$$

Thus, $\left(-2,\frac{7\pi }{3}\right)$ is the required point where $r\ne 2$ but $\theta \ne \frac{\pi }{3}$.

Therefore, the answer is $\left(-2,\frac{7\pi }{3}\right)$.