Q. 2 - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Two pairs of polar coordinates for the point $(0,3)$ given in rectangular coordinates.

(b) Two equations for the line $y=x$ in polar coordinates.

(c) The equations of two distinct circles with radius 2 tangent to the the $x$-axis at the pole in polar coordinates.

Step-by-Step Solution

Answer

(a). The two pair of angles are $\left(3,\frac{\pi }{2}\right)\left(-3,\frac{3\pi }{2}\right).$

(b). The required polar coordinate form of the equation is $\theta =\frac{\pi }{4}.$

(c).  The equation is $r=4\cos \theta .$

Part(a) Step 1: Given information

The coordinate $(0,3), x=0$ and $y=3$.

Part(a) Step 2: Calculation

The objective is to convert the rectangular coordinates to polar coordinates.

In the coordinate $(0,3), x=0$and $y=3.$

To find the value of $r$ use the equation $r=\sqrt{x^2+y^2}.$

Then,

$$\begin{gathered} r=\sqrt{(0{)}^2+(3{)}^2}[sincex=0,y=3] \\ r=\sqrt{9} \\ r=\pm 3 \end{gathered}$$

To calculate $\theta$ use the formula $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$.

By substituting $x, y$in the formula,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{3}{0}\right)[sincex=0,y=3] \\ \theta ={\tan }^{-1}(\infty ). \\ \theta =\frac{\pi }{2},\frac{3\pi }{2}\left[since\tan \frac{\pi }{2}=\tan \frac{3\pi }{2}=\infty \right] \end{gathered}$$

Take$r=3,\theta =\frac{\pi }{2}.$

Then $(r,\theta )=\left(3,\frac{\pi }{2}\right).$

Take $r=-3,\theta =\frac{\pi }{2}$

$(r,\theta )=\left(3,\frac{\pi }{2}\right)\left(-3,\frac{3\pi }{2}\right)$

Therefore the two pairs of angles are $\left(3,\frac{\pi }{2}\right)\mathrm{and}\left(-3,\frac{3\pi }{2}\right)$. 

Part(b) Step 1: Given information

Two equations for the line y=x in polar coordinates.

Part(b) Step 2: Calculation

Consider the equation in rectangular coordinates $y=x$.

The objective is to convert the equation in rectangular coordinates to polar coordinates.

Take the equation $y=x$

The values of $x, y$ in polar coordinates is $x=r\cos \theta ,y=r\sin \theta$.

Then by equating the values of the equation,

$r\cos \theta =r \sin \left(\theta \right)$[ since the equation is $x=y$]

Divide by ron both sides of the equation.

$$\begin{gathered} \frac{r\cos \theta }{r}=\frac{r\sin \theta }{r} \\ \cos \theta =\sin \theta \end{gathered}$$

Divide by $\sin \theta$ on both sides of the equation.

$$\begin{gathered} \frac{\cos \theta }{\sin \theta }=\frac{\sin \theta }{\sin \theta } \\ \cot \theta =1 \\ \theta =\frac{\pi }{4}\left[\text{ since }\cot \frac{\pi }{4}=1\right] \end{gathered}$$

The polar coordinate form of the equation $y=x$ is $\theta =\frac{\pi }{4}$.

Therefore, the required polar coordinate form of the equation is $\theta =\frac{\pi }{4}$.

Part(c) Step 1: Given information

The equations of two distinct circles with radius 2 tangent to the the x-axis at the pole in polar coordinates.

Part(c) Step 2: Calculation

The equation of a circle with radius $r=2$.

The equation $r=4\cos \theta$ is a circle with radius 2 .

Therefore, the equation is $r=4\cos \theta$.