Polar Coordinates

Polar Coordinates Let P$(x,y)$ be a point on a plane. Then we represent P as a polar coordinate. Let the origin O be a pole. Then consider ray from the origin. In this case, $x$-axis is called a polar axis and the angle $\theta$ formed by ray and polar axis is called argument. Let $\vert r\vert$ be the distance between the origin and a point P. Then the position of P can be expressed as a pair such as $[r, \theta]$. Now we call this pair $[r, \theta]$ polar coordinate of P.

$[r, \theta]$ Note that $r$ can be negative.

NOTE Let a rectangular coordinate of P be $(x,y)$ and a polar coordinate of P be $[r, \theta]$. Then we have

$$x = r\cos{\theta}, y = r\sin{\theta}$$

Thus,

$$\tan{\theta} = \frac{y}{x} , r^2 = x^2 + y^2$$

Then given $r$ and $\theta$, P$(x,y)$ is fixed. On the other hand, even P$(x,y)$ is given, the value of $r$ and $\theta$ can not be determined uniquely.

Example 2..20   Find a polar coordinats of the point P that has rectangular coordinate $(1,\sqrt{3})$.

Figure 2.14: Example2-20

SOLUTION Since $x = 1, y = \sqrt{3}$, $\tan{\theta} = \sqrt{3}, r^2 = 1 + 3 = 4$. Thus the point P is on the ray $${\theta = \frac{\pi}{3}}$$ and the distance from the origin is 2. Thus the rectangular coordinate of P can be expressed as $${[2,\frac{\pi}{3}]}$$. Note that $${\theta = \frac{4\pi}{3}}$$ and $r = -2$ also represents the point P. $\blacksquare$

Exercise 2..20   Find the rectangular coordinates of the point P with polar coordinates $[1,\frac{\pi}{3}]$ .

SOLUTION $r = 1, \theta = \frac{\pi}{3}$

$$x = r\cos{\theta} = \cos{\frac{\pi}{3}} = \frac{1}{2}$$

$$y = r\sin{\theta} = \sin{\frac{\pi}{3}} = \frac{\sqrt{3}}{2}\ensuremath{ \blacksquare}$$

Polar Equation Suppose the curve of a function is given by rectangular coordinates $g(x,y) = 0$. Then the equation expressed by the polar coordinates ,

$$g(r\cos{\theta}, r\sin{\theta}) = 0$$

is called Polar Equation.

Example 2..21   Write the equation in polar coordinates

$$x^2 + y^2 = 4$$

Setting $x = r\cos{\theta}, y = r\sin{\theta}$ and eliminate $x,y$.

SOLUTION Let $x = r\cos{\theta}, y = r\sin{\theta}$. Then we have $r^2 = x^2 + y^2 = 4$ which implies $r = 2, -2$ $\blacksquare$

Exercise 2..21   Write the equation in polar coordinates

$$(x-1)^2 + y^2 = 1$$

SOLUTION Let $x = r\cos{\theta}, y = r\sin{\theta}$. Then $(r\cos{\theta} - 1)^2 + (r\sin{\theta})^2 = 1$ $r^2 \cos^{2}{\theta} - 2r\cos{\theta} + 1 + r^2 \sin^{2}{\theta} = 1.$ Simplifying, $r^2 - 2r\cos{\theta} = r(r - 2\cos{\theta}) = 0.$ From this, $r = 2\cos{\theta}$ $\blacksquare$

Example 2..22   Sketch the following polar curve

$$r(\theta) = 1 + \cos{\theta}$$

Figure 2.15: Example2-22

If $r(-\theta) = r(\theta)$, then for positive $\theta$ and negative $\theta$ give the same $r$. Thus, the curve is symmetric with respect to the $x$-axis.

SOLUTION 1. Since $\cos{\theta}$ is a even function, we have $r(-\theta) = r(\theta)$. Thus it is symmetric with respect to the $x$-axis. Thus to draw the curve of a function, we only need to check from $\theta = 0$ to $\theta = \pi$.
2. Next we write a table for a polar coordinates of a curve.

$$\begin{array}{c\vert ccccccc} \theta & 0 & \frac{\pi}{6} & \... ... 1 & 1 - \frac{1}{2} & 1 - \frac{\sqrt{3}}{2} & 0 \end{array}$$

From this, we have Cardioid $\blacksquare$

Exercise 2..22   Sketch the following polar curve

$$r(\theta) = 1 + 2\cos{\theta}$$

Figure 2.16: Exercise2-22

SOLUTION

1. $\cos{\theta}$ is even implies $r(-\theta) = r(\theta)$. Thus it's curve is symmetric with respect ot the $x$-axis.
2. Next we write a table for a polar coordinates of a curve.
   $$\begin{array}{c\vert ccccccc} \theta & 0 & \frac{\pi}{6} & \... ... + \sqrt{3} & 1 + 1 & 1 & 1- 1 & 1- \sqrt{3} & -1 \end{array}$$

The curve is called Limason $\blacksquare$

Exercise A

1.

Find the critical points of $f$ and the local extreme values. Describe the concavity of the graph of $f$ and find the points of inflection.

(a) $${f(x) = x^{3} - 3x + 2}$$ (b) $${f(x) = x + \frac{1}{x}}$$ (c) $${f(x) = x(x+1)(x+2)}$$

(d) $${f(x) = \frac{x}{1+x^{2}}}$$ (e) $${f(x) = \vert x-1\vert\vert x+2\vert}$$

2.

Find the asymtote of the following functions

(a) $${f(x) = \frac{x}{3x-1}}$$ (b) $${y = \frac{x^{2}}{x-2}}$$ (c) $${y = \frac{2x}{x^{2}-9}}$$

2.

Determine the following function has a vertical cuspWhen a function satisfies the condition $f'(c-0) = \pm \infty$ and $f'(c+0) = \mp \infty$, we say $(c,f(c)$ is a vertical cusp

(a) $${f(x) = x^{1/3}}$$ (b) $${f(x) = x^{2/3}}$$ (c) $${f(x) = \sqrt{\vert x-2\vert}}$$

Exercise B

4.

Find the local extreme value of $f$ and describe the concavity of the graph of $f$

(a) $${f(x) = x^{3} - 6x^2 + 9x + 3}$$ (b) $${f(x) = x^{2}e^{-x}}$$

1.

Sketch the graph

(a) $${y = \frac{1}{1 + e^{x}}}$$ (b) $${y = \frac{1-x}{1 + x}}$$

2.

Sketch the graph

(a) $${r = a\cos{\theta}, a > 0}$$ (circle)

(b) $${r = 3\theta}$$, Archmedes' spiral

(c) $${r^2 = 4\cos{2\theta}}$$ Bernoulli's lemniscate