Polar Coordinates

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

• Section 10.3 – Polar Coordinates

Expected Educational Results

• Objective 16–01: I can convert a point between polar coordinates and cartesian coordinates.

• Objective 16–02: I can convert equations between polar coordinates and cartesian coordinates.

• Objective 16–03: I can use technology to sketch a polar curve.

• Objective 16–04: I can find the derivative of a polar curve.

• Objective 16–05: I can use the derivative of a polar curve to find the equation of the tangent line to the curve, points where the curve has horizontal tangent lines, and points where the curve has vertical tangent lines.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Polar Coordinate System

Definition: Derivatives of Polar Curves

Let $r$$r$ be a function of $\theta$$\theta$, i.e., $r(\theta )$$r(\theta)$. We know $x=r\cos (\theta )$$x=r\cos{(\theta)}$ and $y=r\sin (\theta )$$y=r\sin{(\theta)}$. Then

${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{{\displaystyle \frac{dy}{d\theta }}}{{\displaystyle \frac{dx}{d\theta }}}}={\displaystyle \frac{{\displaystyle \frac{dr}{d\theta }}\sin (\theta )+r\cos (\theta )}{{\displaystyle \frac{dr}{d\theta }}\cos (\theta )-r\sin (\theta )}}$$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{d\theta}}{\dfrac{dx}{d\theta}}=\dfrac{\dfrac{dr}{d\theta}\sin{(\theta)}+r\cos{(\theta)}}{\dfrac{dr}{d\theta}\cos{(\theta)}-r\sin{(\theta)}}$

Derivation

NOTE: You will not be assessed on the derivation of the derivative of a polar curve.

Numerator: ${\displaystyle \frac{dy}{d\theta }}={\displaystyle \frac{d}{d\theta }}(r(\theta )\sin (\theta ))$$\dfrac{dy}{d\theta}=\dfrac{d}{d\theta}\left(r(\theta)\sin{(\theta)}\right)$

$\Rightarrow ({\displaystyle \frac{d}{d\theta }}r(\theta ))\sin (\theta )+r(\theta )({\displaystyle \frac{d}{d\theta }}\sin (\theta ))$$\Rightarrow\left(\dfrac{d}{d\theta}r(\theta)\right)\sin{(\theta)}+r(\theta)\left(\dfrac{d}{d\theta}\sin{(\theta)}\right)$ by Product Rule for derivatives

$\Rightarrow \frac{dr}{d\theta }\sin (\theta )+r\cos (\theta )$$\Rightarrow\frac{dr}{d\theta}\sin{(\theta)}+r\cos{(\theta)}$

Denominator: ${\displaystyle \frac{dx}{d\theta }}={\displaystyle \frac{d}{d\theta }}(r(\theta )\cos (\theta ))$$\dfrac{dx}{d\theta}=\dfrac{d}{d\theta}\left(r(\theta)\cos{(\theta)}\right)$

$\Rightarrow ({\displaystyle \frac{d}{d\theta }}r(\theta ))\cos (\theta )+r(\theta )({\displaystyle \frac{d}{d\theta }}\cos (\theta ))$$\Rightarrow\left(\dfrac{d}{d\theta}r(\theta)\right)\cos{(\theta)}+r(\theta)\left(\dfrac{d}{d\theta}\cos{(\theta)}\right)$ by Product Rule for derivatives.

$\Rightarrow {\displaystyle \frac{dr}{d\theta }}\cos (\theta )+r(-\sin (\theta ))$$\Rightarrow\dfrac{dr}{d\theta}\cos{(\theta)}+r\left(-\sin{(\theta)}\right)$

$\Rightarrow {\displaystyle \frac{dr}{d\theta }}\cos (\theta )-r\sin (\theta )$$\Rightarrow\dfrac{dr}{d\theta}\cos{(\theta)}-r\sin{(\theta)}$

Horizontal Tangent Lines to Polar Curves

Corollary

A polar curve has horizontal tangent line(s) where ${\displaystyle \frac{dr}{d\theta }\sin (\theta )+r\cos (\theta )=0}$$\displaystyle\frac{dr}{d\theta}\sin{(\theta)}+r\cos{(\theta)}=0$ and ${\displaystyle \frac{dx}{d\theta }\ne 0}$$\displaystyle\frac{dx}{d\theta}\ne 0$

Vertical Tangent Lines to Polar Curves

Corollary

A polar curve has vertical tangent line(s) where ${\displaystyle \frac{dr}{d\theta }\cos (\theta )-r\sin (\theta )=0}$$\displaystyle\frac{dr}{d\theta}\cos{(\theta)}-r\sin{(\theta)}=0$ and ${\displaystyle \frac{dy}{d\theta }\ne 0}$$\displaystyle\frac{dy}{d\theta}\ne 0$

Investigation 08

1. Find the equation of the tangent line to ${\displaystyle r=3\cos (\theta )}$$\displaystyle r=3\cos{(\theta)}$ at ${\displaystyle \theta =\frac{\pi }{3}}$$\displaystyle \theta=\frac{\pi}{3}$.

2. Find the location(s) [in terms of $\theta$$\theta$ and on the interval $0\le \theta <2\pi$$0\leq\theta<2\pi$] where ${\displaystyle r=3\cos (\theta )}$$\displaystyle r=3\cos{(\theta)}$ has horizontal tangent lines.

3. Find the location(s) [in terms of $\theta$$\theta$ and on the interval $0\le \theta <2\pi$$0\leq\theta<2\pi$] where ${\displaystyle r=3\cos (\theta )}$$\displaystyle r=3\cos{(\theta)}$ has vertical tangent lines.

Use Technology to Solve Equations Exactly

NOTE: On all Assessments, you may use technology to solve equations without showing any work.

Mathematica

​x
1
(* Example from Investigation 08-2 *)
2
(* Use the variable t instead of theta *)
3

4
(* Find locations of horizontal tangent lines to r=3cos(t) *)
5
Solve[-3Sin[t]Sin[t]+3Cos[t]Cos[t]==0, t, Reals]
6

7
(* Restrict answers to [0,2pi) *)
8
Solve[-3Sin[t]Sin[t]+3Cos[t]Cos[t]==0 && 0 <= t < 2 Pi, t, Reals]
9

10
(* If the answers from the above code are not easy to interpret *)
11

12
(* Simplify the trig equation *)
13
TrigReduce[-3 Sin[t] Sin[t] + 3 Cos[t] Cos[t]]
14

15
(* Solve using the simplified trig equation *)
16
Solve[3 Cos[2 t] == 0 && 0 <= t < 2 Pi, t, Reals]

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. Mathematica requires two $==$$==$ for the equals sign in an equation.

3. Solve[ ] has at least two arguments:

   1. The equation to be solved;

      1. $\text{\&\&}$$\text{\&\&}$ is Mathematica code for “and,” i.e., both equation and condition must be true

   2. The variable, i.e., $t$$t$, to be solved for;

   3. The argument $\text{Reals}$$\text{Reals}$ to force Mathematica to return only real-valued results.

4. Mathematica may return complex solutions; thus, either

   1. Identify a complex solution (the result includes the imaginary unit, $i$$i$), or

   2. Use the argument Reals.

5. For help on using the Solve[ ] function:

   1. In Mathematica, execute the code: $\text{?Solve}$$\text{?Solve}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Solve[ ] function – you will be able to copy-paste code.

6. Remember, correct Mathematica code will be all black except for variables.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Investigation 09

1. Find the equation of the tangent line to ${\displaystyle r=\sin (3\theta )}$$\displaystyle r=\sin{(3\theta)}$.

2. Estimate (to four decimal places) the location(s) [in terms of $\theta$$\theta$] where ${\displaystyle r=\sin (3\theta )}$$\displaystyle r=\sin{(3\theta)}$ has horizontal tangent lines.

3. Estimate (to four decimal places) the location(s) [in terms of $\theta$$\theta$] where ${\displaystyle r=\sin (3\theta )}$$\displaystyle r=\sin{(3\theta)}$ has vertical tangent lines.

Use Technology to Approximately Solve Equations

NOTE: On all Assessments, you may use technology to solve equations without showing any work.

Mathematica

xxxxxxxxxx
8
1
(* Example from Investigation 09-2 *)
2
(* Use the variable t instead of theta *)
3

4
(* Find approx locations of horizontal tangent lines to r=sin(3t) *)
5
Solve[3Cos[3t]Sin[t]+Sin[3t]Cos[t]==0, t, Reals]//N
6

7
(* Restrict answers to [0,pi) *)
8
Solve[3Cos[3t]Sin[t]+Sin[3t]Cos[t]==0 && 0 <= t < Pi, t, Reals]//N

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. Mathematica requires two $==$$==$ for the equals sign in an equation.

3. Solve[ ] has at least two arguments:

   1. The equation to be solved;

      1. $\text{\&\&}$$\text{\&\&}$ is Mathematica code for “and,” i.e., both equation and condition must be true

   2. The variable, i.e., $t$$t$, to be solved for;

   3. The argument $\text{Reals}$$\text{Reals}$ to force Mathematica to return only real-valued results.

4. Mathematica may return complex solutions; thus, either

   1. Identify a complex solution (the result includes the imaginary unit, $i$$i$), or

   2. Use the argument Reals.

5. $\text{//N}$$\text{//N}$ at the end of a line of code results in a decimal approximation

6. For help on using the Solve[ ] function:

   1. In Mathematica, execute the code: $\text{?Solve}$$\text{?Solve}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Solve[ ] function – you will be able to copy-paste code.

7. Remember, correct Mathematica code will be all black except for variables.

8. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

At this time, you should be able to complete the following assignments:

• Section 10.3: # 55, 57, 59, 61, 63.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 3 November 2020 6:52 EDT Last Modified: Thursday, 14 July 2022 - 18:04 (EDT)