Polar Coordinates

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 10.3 – Polar Coordinates

Polar Coordinate System

Prerequisite Knowledge

This section contains the specific content from previous courses that you may need for Polar Coordinates.

Trigonometric Ratios on Right Triangle

• ${\displaystyle \cos (\theta )=\frac{\text{length of adjacent side}}{\text{length of hypotenuse}}}$$\displaystyle \cos{(\theta)}=\frac{\text{length of adjacent side}}{\text{length of hypotenuse}}$
• ${\displaystyle \sin (\theta )=\frac{\text{length of opposite side}}{\text{length of hypotenuse}}}$$\displaystyle \sin{(\theta)}=\frac{\text{length of opposite side}}{\text{length of hypotenuse}}$

Trigonometric Ratio on Unit Circle

• ${\displaystyle \tan (\theta )=\frac{y}{x}}$$\displaystyle \tan{(\theta)}=\frac{y}{x}$

Angles and Reference Angles

Let ${\displaystyle {\theta }_{\text{ref}}}$$\displaystyle \theta_{\text{ref}}$ be the reference angle formed with the $x$$x$-axis. ${\displaystyle {\theta }_{\text{ref}}=0}$$\displaystyle \theta_{\text{ref}}=0$, ${\displaystyle \frac{\pi }{6}}$$\displaystyle\frac{\pi}{6}$, ${\displaystyle \frac{\pi }{4}}$$\displaystyle\frac{\pi}{4}$, ${\displaystyle \frac{\pi }{3}}$$\displaystyle\frac{\pi}{3}$, or ${\displaystyle \frac{\pi }{2}}$$\displaystyle\frac{\pi}{2}$.

Let $\theta$$\theta$ be the angle formed with the positive $x$$x$-axis in the counter-clockwise direction.

Computation of Angles

According to Trigonometry,

• angles, $\theta$$\theta$, are formed from the positive $x$$x$-axis using a counter-clockwise orientation;
• reference angles, ${\theta }_{\text{ref}}$$\theta_{\text{ref}}$, are formed with the $x$$x$-axis.

To compute an angle, where $0\le \theta <2\pi$$0\leq\theta< 2\pi$ (NOTE: $\theta =0$$\theta=0$ is equivalent [not equal] to $\theta =2\pi$$\theta=2\pi$):

1. Find the reference angle, ${\theta }_{\text{ref}}$$\theta_{\text{ref}}$, i.e., the angle in Quadrant I, on the positive $x$$x$-axis, or on the positive $y$$y$-axis:

   • ${\displaystyle {\theta }_{\text{ref}}={\tan }^{-1}\left(\frac{y}{x}\right)}$$\displaystyle\theta_{\text{ref}}=\tan^{-1}{\!\left(\frac{y}{x}\right)}$, where $y\ge 0$$y\geq 0$ and $x>0$$x> 0$

2. Use the Quadrant where the point $(x,y)$$(x,y)$ is located to find $\theta$$\theta$:

   • $x$$x$	$y$$y$	Quadrant	$\theta$$\theta$
     $x>0$$x>0$	$y>0$$y > 0$	I	$\theta ={\theta }_{\text{ref}}$$\theta=\theta_{\text{ref}}$
     $x<0$$x<0$	$y>0$$y>0$	II	$\theta =\pi -{\theta }_{\text{ref}}$$\theta=\pi-\theta_{\text{ref}}$
     $x<0$$x<0$	$y<0$$y<0$	III	$\theta =\pi +{\theta }_{\text{ref}}$$\theta=\pi+\theta_{\text{ref}}$
     $x>0$$x>0$	$y<0$$y<0$	IV	$\theta =2\pi -{\theta }_{\text{ref}}$$\theta=2\pi-\theta_{\text{ref}}$

Half-angle formulas

• ${\displaystyle {\sin }^2(x)=\frac{1-\cos (2x)}{2}}$$\displaystyle \sin^2{(x)}=\frac{1-\cos{(2x)}}{2}$
• ${\displaystyle {\cos }^2(x)=\frac{1+\cos (2x)}{2}}$$\displaystyle \cos^2{(x)}=\frac{1+\cos{(2x)}}{2}$

Use Technology to Recall Half-Angle Formulas

Mathematica

1
(* Example: Rewrite [cos(x)]^2 *)
2
TrigReduce[Cos[x]^2]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.
2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
3. Mathematica may return a result in a different, but equivalent, form than your answer.

Double-Angle Formulas

• $\sin (2x)=2\sin (x)\cos (x)$$\sin{(2x)}=2\sin{(x)}\cos{(x)}$
• $\cos (2x)={\cos }^2(x)-{\sin }^2(x)$$\cos{(2x)}=\cos^2{(x)}-\sin^2{(x)}$

Use Technology to Recall Double-Angle Formulas

Mathematica

xxxxxxxxxx
2
1
(* Example: Rewrite cos(2x) *)
2
TrigExpand[Cos[2x]]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.
2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
3. Mathematica may return a result in a different, but equivalent, form than your answer.

Evaluation of Trig Functions

There is NO reason to memorize the unit circle. Look at the Table below and consider the patterns of the numbers in red. Use these patterns to recall your trig ratios.

Rational Functions

Rational functions have the form ${\displaystyle R(x)=\frac{P(x)}{Q(x)}}$$\displaystyle R(x)=\frac{P(x)}{Q(x)}$.

Horizontal Asymptotes of Rational Functions

$R(x)$$R(x)$ has a horizontal asymptote when $P(x)=0$$P(x)=0$ and $Q(x)\ne 0$$Q(x)\ne 0$.

Vertical Asymptotes of Rational Functions

$R(x)$$R(x)$ has a vertical asymptote when $Q(x)=0$$Q(x)=0$ and $P(x)\ne 0$$P(x)\ne 0$.