Important information about the FINAL EXAM

Hi everyone,

The Mathematics Department has released information about how the final exams will be conducted this semester. I want to give you a quick overview, and share the more detailed information from the department.

How will you take the final exam?

The process for the final exam will be similar to our Exam #2:
1. On the day of the exam, I will post Exam information on the OpenLab
2. You will complete the exam in WeBWorK using a special “Exam WeBWorK account”, NOT your usual WeBWorK account – more info on this later. You will have a 48-hour window to start the exam, and a 2-hour window to complete the exam once you start it.
3. You will upload your written work in pdf format to a dropbox link that I will provide.

What is my “Exam WeBWorK Account”?

This is a special WeBWorK account created just for you to take the final exam (and the practice final exam). It is NOT the same as the WeBWorK account you have been using all semester.

You should have received an email sometime today (Friday, 5/8/20) with the title Practice Exams and Final Exam Login Information from the address WeBWorK Administrator <webwork@citytech.cuny.edu>. This email contains your exam WeBWorK account login and password, and a link to the WeBWorK final exam area for our class.

If you didn’t receive your login/password information today, please let me know AS SOON AS POSSIBLE!

Please read the information provided by the Math Department – Frequently Asked Questions for students:

Important information about your grade (CUNY Credit/No Credit Policy)

Hi everyone,

This is indeed a very unusual semester at CUNY – I thank you all for sticking with it, despite the challenges!

I want to make you aware of a policy created by the central CUNY administration that may help you with your grades this semester. Take a look at the info and resources below, and please let me know if you have any questions.

What is the CUNY Credit/No Credit option?
For this semester only, in each of your classes you will be able to choose between
a) receiving a letter grade as usual (A,B,C,D,F, etc.), and
b) receiving a “Credit / No Credit” grade (like a pass/fail).

Why would I want to choose the Credit / No Credit option?
If you choose the Credit/No Credit option, your grade will not affect your GPA. If you pass the class, you will receive a CR (credit) grade, if you don’t pass the class you will receive a NC (no credit) grade. This is not the only benefit, and there are possible drawbacks as well – please read the policy faq carefully before you decide.

Why would I want to choose the letter grade option?
If you choose the letter grade option, you grade will affect your GPA. This is not the only benefit, and there are possible drawbacks as well – please read the policy faq carefully before you decide.

When do I have to make my decision?
You will be able to make the decision (in CUNYFirst) after the final grades have been entered by your instructor, but before June 25th. Basically, you can wait until you receive you letter grade at the end of the semester, and then decide whether you want to convert it to a credit/no credit grade.

info

CUNY Credit / No Credit Policy info

1. CUNY’s Frequently Asked Questions page about the policy.
2. City Tech’s info page about the policy.

Lesson 23: Sequences and series

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson 23: Sequences and series

Lesson Date: Thursday, May 7th.

Topic: This lesson covers Chapter 23: Sequences and series

WeBWorK: There are three WeBWorK assignments on today’s material, due in one week:

Sequences – Introduction

Sequences – Arithmetic

Series – Finite Arithmetic

Question of the day: What is a sequence? What is a series?

Answer of the day: A sequence is just a list of numbers. A series is list of numbers, added up.

Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Introduction to sequences and series

Today we will introduce two very powerful ideas (they are the building blocks of Calculus) – however, the ideas themselves are not complicated: sequences and series. They both have to do with lists of numbers, rather than individual numbers.

Definition 23.1. A sequence is an ordered list of numbers (we call each number in the list a term in the sequence). We write a sequence in order as follows: a_{1}, a_{2}, a_{3}, a_{4},...
In short we write the above sequence as \left\{a_{n}\right\} or \left\{a_{n}\right\}_{n \geq 1}

Let’s look at examples of some sequences. As you look at this example, keep the following questions in mind:

  • Is there a pattern in the sequence?
  • Can you predict what the next term of the sequence will be?
  • Can you predict what the 10th term of the sequence will be? What about the 20th, or the 100th?
  • Can you find a formula for a_n, the nth term of the sequence?

Example 23.2. Here are some examples of sequences.
a) 4,6,8,10,12,14,16,18, \dots
b) 1,3,9,27,81,243, \dots
c) +5,-5,+5,-5,+5,-5, \dots
d) 1,1,2,3,5,8,13,21,34,55, \dots
e) 5,8,-12,4,5.3,7,-3, \sqrt{2}, 18, \frac{2}{3}, 9, \dots

Example 23.3. Consider the sequence \left\{a_{n}\right\} with a_{n}=4 n+3. Calculate the first five terms of the sequence a_1, a_2, a_3, a_4, and a_5. What is the 200th term of the sequence?

VIDEO: Introduction to sequences – Examples 23.2 and 23.3

Example 23.4. Find the first 6 terms of each sequence.
a) a_{n}=n^{2}
b) a_{n}=\frac{n}{n+1}
c) a_{n}=(-1)^{n}
d) a_{n}=(-1)^{n+1} \cdot 2^{n}

Another way to describe a sequence is by giving a recursive formula for the nth term a_{n} in terms of the lower terms. Here are some examples.

Example 23.6. Find the first 6 terms in the sequence described below.
a) a_{1}=4, and a_{n}=a_{n-1}+5 for n>1
b) a_{1}=3, and a_{n}=2 \cdot a_{n-1} for n>1
c) a_{1}=1, a_{2}=1, and a_{n}=a_{n-1}+a_{n-2} for n>2

VIDEO: Two ways of describing sequences -Examples 23.4 and 23.6

Definition 23.8. A series is a sum of terms in a sequence. We denote the sum of the first k terms in a sequence with the following notation:
\sum_{i=1}^{k} a_{i}=a_{1}+a_{2}+\cdots+a_{k}
The summation symbol \sum comes from the greek letter \Sigma, pronounced “sigma,” which is the greek letter for “S.”

Example 23.9. Find the sum.
a) \sum_{i=1}^{4} a_{i}, for a_{i}=7 i+3
b) \sum_{j=1}^{6} a_{j}, for a_{n}=(-2)^{n}
c) \sum_{k=1}^{5}\left(4+k^{2}\right)

VIDEO: Intro to series – Example 23.9

The arithmetic sequence

Definition 23.10. A sequence \left{a_{n}\right} is called an arithmetic sequence if any two consecutive terms have a common difference d. The arithmetic sequence is determined by d and the first value a_{1}. This can be written recursively as:
a_{n}=a_{n-1}+d \quad \text { for } n \geq 2
Alternatively, we have the general formula for the nth term of the arithmetic sequence
a_{n}=a_{1}+d \cdot(n-1)

Example 23.11. Determine if the sequence is an arithmetic sequence. If so, then find the general formula for a_{n}.
a) 7,13,19,25,31, \ldots
b) 13,9,5,1,-3,-7, \dots
c) 10,13,16,20,23, \dots
d) a_{n}=8 \cdot n+3

VIDEO: Arithmetic sequences – Example 23.11

Example 23.12. Find the general formula of an arithmetic sequence with the given property.
a) d=12, and a_{6}=68
b) a_{1}=-5, and a_{9}=27
c) a_{5}=38, and a_{16}=115

VIDEO: Arithmetic sequences – finding a general formula, Example 23.12

Example 23.13. Find the sum of the first 100 integers, starting from 1 . In other words, we want to find the sum of 1+2+3+\cdots+99+100.

VIDEO: Arithmetic series – intro example 23.13

Observation 23.14 . Let \left\{a_{n}\right\} be an arithmetic sequence, whose n th term is given by the formula a_{n}=a_{1}+d(n-1). Then, the sum a_{1}+a_{2}+\cdots+a_{k-1}+a_{k} is given by adding \left(a_{1}+a_{k}\right) precisely \frac{k}{2} times:
\sum_{i=1}^{k} a_{i}=\frac{k}{2} \cdot\left(a_{1}+a_{k}\right)

Example 23.15. Find the value of the arithmetic series.
a) Find the sum a_{1}+\cdots+a_{60} for the arithmetic sequence a_{n}=2+13(n-1)
b) Determine the value of the sum: \quad \sum_{j=1}^{1001}(5-6 j)
c) Find the sum of the first 35 terms of the sequence
4,3.5,3,2.5,2,1.5, \ldots

VIDEO: Arithmetic series – finding the sum, Example 23.15

Daily Quiz and Attendance: 5/7/20

Hi everyone,

Here is today’s Daily Quiz, based on the material covered on Tuesday.   As a reminder, the quizzes will count towards the OpenLab portion of your grade, and they will also be used to track your attendance (to be marked PRESENT, you must submit the quiz before midnight). Today’s lecture will be posted later this morning.

As always, let me know if you have any questions or problems. 

Be well,
Prof.  Reitz

Daily Quiz: 5/7/20

This quiz MUST be taken on 5/7 to record your attendance.

Exam 2 Grades are posted

Hi everyone,

I’ve finished grading exam 2. This is our first “distance” exam, so I request your patience in navigating various technological hurdles – please read the following important messages before you check your grade!

  • This exam consisted of 6 problems, each worth 17 points (that means a perfect score would earn you a grade of 102 – a little bonus).
  • You can view your grade here on the OpenLab under Dashboard/OpenLab GradeBook (or look for the Gradebook widget on the right side of the screen)
  • In the gradebook you should see a little blue “comment” bubble next to your grade – this contains a link to your graded exam paper (pdf).
  • If your name does not appear in the OpenLab Gradebook, you need to register for the OpenLab and join the course! (then let me know, and I will add your grade information)
  • If you took the exam but did not receive a grade, please send me an email and we will track down the problem.

All the best,
Prof. Reitz

Lesson 22: Vectors in the plane

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson 22: Vectors in the plane

Lesson Date: Tuesday, May 5th.

Topic: This lesson covers Chapter 22: Vectors in the plane.

WeBWorK: There are four WeBWorK assignments on today’s material, due next Tuesday 5/12:

Vectors – Components

Vectors – Operations

Vectors – Magnitude and Direction

Vectors – Unit Vectors

Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Introduction to Vectors

Today we will be working with the plane \mathbb{R}^2, but looking at things in a slightly different way – instead of points (which have only a location), we will be focussing on vectors (which have a magnitude (size) and direction). This change in perspective is quite powerful, and brings to light many useful features of the plane – but in practice, you will find it similar to the work we did in the previous lesson on polar form of complex numbers.

Definition 22.1. A geometric vector in the plane is a geometric object in the plane \mathbb{R}^{2} that is given by a direction (angle) and magnitude (size). We denote a vector by \vec{v} (it is written by some authors as v ), its magnitude is denoted by ||\vec{v}||, and its directional angle by \theta.

Vectors are often drawn as directed line segments \vec{v}=\overrightarrow{P Q}. Two such segments represent the same vector if they have the same magnitude and direction.

Four different drawings of the vector v
Four different drawings of the vector \vec{v}.

We can always represent a vector \vec{v} by arranging the starting point of \vec{v} to be the origin O(0,0) (as in \overrightarrow{O R} in the picture above). If R has coordinates R(a, b) then we also write for \vec{v}=\overrightarrow{O R}:
\vec{v}=\langle a, b\rangle, or \vec{v}=\begin{bmatrix} a\\ b\end{bmatrix}

Example 22.2. Graph the vectors \vec{v}, \vec{w}, \vec{r}, \vec{s}, \vec{t} in the plane, where
\vec{v}=\overrightarrow{P Q} with P(6,3) and Q(4,-2), and
\vec{w}=\langle 3,-1\rangle, \quad \vec{r}=\langle-4,-2\rangle, \quad \vec{s}=\langle 0,2\rangle, \quad \vec{t}=\langle-5,3\rangle

Solution - Example 22.2
Solution – Example 22.2

The formulas for magnitude and directional angle of a vector are the same as those for modulus (magnitude) and argument (angle) of a complex number:

Formulas for magnitude and angle of a vector: Suppose \vec{v}=\langle{a,b}\rangle} is a vector in the plane \mathbb{R}^2. Then the magnitude and angle of \vec{v} are given by:

||\vec{v}||=\sqrt{a^2+b^2}, \tan(\theta)=\frac{b}{a}

Conversely, we can obtain the coordinates of a vector from its magnitude and directional angle by:

\vec{v}=\langle||\vec{v}|| \cdot \cos (\theta),||\vec{v}|| \cdot \sin (\theta)\rangle

Example 22.4. Find the magnitude and directional angle of the given vectors:
a) \langle-6,6\rangle
b) \langle 4,-3\rangle
c) \langle-2 \sqrt{3},-2\rangle
d) \langle 8,4 \sqrt{5}\rangle
e) \overrightarrow{P Q}, where P(9,2) and Q(3,10)

VIDEO: Intro to vectors, finding magnitude and direction – Example 22.4

Operations on Vectors

There are two basic operations on vectors, scalar multiplication and vector addition.

Scalar Multiplication

Definition 22.5. The scalar multiplication of a real number r with a vector \vec{v}=\langle a, b\rangle is defined to be the vector given by multiplying each coordinate by r:
r \langle a, b\rangle =\langle r \cdot a, r \cdot b\rangle

Example 22.6. Multiply, and graph the vectors
a) 4 \cdot\langle-2,1\rangle
b) (-3)\cdot\langle-6,-2\rangle

VIDEO: Scalar multiplication of vectors – Example 22.6

Observation. When we multiply a vector \vec{v} by a positive real number r>0, the result will have the same angle as \vec{v}, while the magnitude will be stretched by a factor of r.

The Unit Vector

Definition 22.8. A vector \vec{u} is called a unit vector if it has a magnitude of 1
\vec{u} is a unit vector \quad \Longleftrightarrow ||\vec{u}||=1

There are two special unit vectors \vec{i} and \vec{j}, which are the vectors pointing in the x- and the y -direction.
\vec{i}=\langle 1,0\rangle \quad \text { and } \quad \vec{j}=\langle 0,1\rangle

Example 22.9. Find a unit vector in the direction of \vec{v}
a) \langle 8,6\rangle
b) \langle-2,3 \sqrt{7}\rangle

VIDEO: Unit vectors – Example 22.9

Vector Addition

The second operation on vectors is called vector addition.

Definition 22.10. Let \vec{v}=\langle a, b\rangle and \vec{w}=\langle c, d\rangle be two vectors. Then the vector addition \vec{v}+\vec{w} is defined by component-wise addition:
\langle a, b\rangle+\langle c, d\rangle:=\langle a+c, b+d\rangle

In the plane, this corresponds to starting at the origin, following \vec{v} and then \vec{w} (or vice versa, following \vec{w} and then \vec{v}). In the picture, note that whichever path you take from the origin you will still arrive at the same point in the upper right, \vec{v}+\vec{w}:

Example 22.11. Perform the vector addition and simplify as much as possible.
a) \langle 3,-5\rangle+\langle 6,4\rangle
b) 5 \cdot\langle-6,2\rangle-7 \cdot\langle 1,-3\rangle
c) 4 \vec{i}+9 \vec{j}
d) find 2 \vec{v}+3 \vec{w} for \vec{v}=-6 \vec{i}-4 \vec{j} and \vec{w}=10 \vec{i}-7 \vec{j}
e) find -3 \vec{v}+5 \vec{w} for \vec{v}=\langle 8, \sqrt{3}\rangle and \vec{w}=\langle 0,4 \sqrt{3}\rangle

VIDEO: Vector addition – Example 22.11

Example 22.12. The forces \vec{F}_{1} and \vec{F}_{2} are applied to an object. Find the resulting total force \vec{F}=\vec{F}_{1}+\vec{F}_{2}. Determine the magnitude and directional angle of the total force \vec{F}. Approximate these values as necessary. Recall that the international system of units for force is the newton \left[1 N=1 \frac{k g \cdot m}{s^{2}}\right]
a) \vec{F}_{1} has magnitude 3 newtons, and angle \theta_{1}=45^{\circ}
\vec{F}_{2} has magnitude 5 newtons, and angle \theta_{2}=135^{\circ}
b) \left|\vec{F}_{1}\right|=7 newtons, and \theta_{1}=\frac{\pi}{6}, and \left|\vec{F}_{2}\right|=4 newtons, and \theta_{2}=\frac{5 \pi}{3}

VIDEO: Vector addition application (force) – Example 22.12

That’s it for now – give the WeBWorK a try and let me know if you have any questions!

Daily Quiz and Attendance: 5/5/20

Hi everyone,

Here is today’s Daily Quiz, based on the material covered last Tuesday.   As a reminder, the quizzes will count towards the OpenLab portion of your grade, and they will also be used to track your attendance (to be marked PRESENT, you must submit the quiz before midnight). Today’s lecture will be posted later this morning.

As always, let me know if you have any questions or problems. 

Be well,
Prof.  Reitz

Daily Quiz: 5/5/20

This quiz MUST be taken on 5/5 to record your attendance.
  • Why don't we just stick with the standard a+bi form?

Exam #2 and Attendance – 4/30/20

Hi everyone,

Please READ the instructions below, and COMPLETE all the steps. Completing the exam will also be used to track your attendance (to be marked PRESENT for today’s class, you must take the exam). If you have any trouble with any of the steps, please reach out to me by email or on the OpenLab – let’s figure it out together!

Exam Instructions – (Exam opens at 10am)

warning

Academic Integrity

You are expected to complete the exam on your own, in the allotted time, without reference to any external websites, notes, people or other resources (you may use a graphing or scientific calculator).

Use of online problem-solving sites or advanced step-by-step calculators during the exam will be considered academic dishonesty and, if detected, will result in a failing grade or other penalties.

Step 1: Complete the exam in WeBWorK.

  • The exam is called Take Reitz Precalc Exam2 test and appears in the Homework Sets area (at the bottom).
  • The exam will be available for just over 24 hours, from 10:00am Thursday 4/30/20 through 11:40am Wednesday 5/1/20 – you can start the exam whenever you like during this period.
  • You will have 2 hours to complete the exam from the time you start it (WeBWorK will keep track of this for you – you must submit your answers within two hours of starting).
  • Please choose a time to complete the exam that provides you with the best opportunity to work uninterrupted.
  • You must complete each of the problems on paper (see Step 2).
  • When you complete the exam, enter your answers and submit them in WeBWorK.

Step 2: Upload your written work as a single pdf file to the link provided.

  • You must submit your written work within 30 minutes of completing the exam.
  • Your exam grade will be primarily based on your written work – so this step is essential!
  • Please take clear photographs of your written work (one photo per page).
  • You MUST combine all the photos into a single pdf file. If you need help with this, see the box “Tips for combining photos on various devices” below.
  • Upload the pdf file by clicking the following link: https://www.dropbox.com/request/pP5X2rQZvFj4ISZngYol

Step 3: You’re done – great work! If you had any trouble carrying out the instructions above, please let me know SOONER rather than later (email is a great way: jreitz@citytech.cuny.edu).

picture_as_pdf

Tips for combining photos in a pdf on various devices


There are many ways to convert photos to pdf documents – if you already have a method that works, great! If not, here are links to a few resources, by device type:
iPhone/iPad: How to save photos as pdf on iPhone and iPad
Android: How to scan documents and photos into PDFs on Android
Mac laptop or desktop: How to Combine Images into One PDF File on a Mac
Windows laptop or desktop: How To Create A PDF From Multiple Images In Windows 10

Note 1: Completing the exam will also mark your attendance for the day.

Note 2: Since we are taking an exam today, there will not be any additional lesson on new material. Finish the exam, upload your work, and then give your brain a break!

Lesson 21: Complex numbers

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson 21: Complex Numbers

Lesson Date: Tuesday, April 28th.

Topic: This lesson covers Chapter 21: Complex numbers.

WeBWorK: There are four WeBWorK assignments on today’s material, due next Thursday 5/5:

Complex Numbers – Operations

Complex Numbers – Magnitude

Complex Numbers – Direction

Complex Numbers – Polar Form

Question of the Day: What is the square root of -1?

Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Review of Complex Numbers

How do we get the complex numbers? We start with the real numbers, and we throw in something that’s missing: the square root of -1.

Definition 21.1. We define the imaginary unit or complex unit to be:

    \[i=\sqrt{-1}\]

The most important property of i is: \quad i^2=-1

Definition 21.2. A complex number is a number of the form a+bi.

a and b are allowed to be any real numbers. a is called the real part of a+bi, and b is called the imaginary part of a+bi. The complex numbers are referred to as \mathbb{C} (just as the real numbers are \mathbb{R}.

We can picture the complex number a+bi as the point with coordinates (a,b) in the complex plane.

Complex numbers represented as points in the plane.
Complex numbers represented as points in the plane.

Example 21.3. Perform the operation.
a) (2-3 i)+(-6+4 i)
b) (3+5 i) \cdot(-7+i)
c) \frac{5+4 i}{3+2 i}

VIDEO: Review of Complex Numbers – Example 21.3

Polar form

Next, we will look at how we can describe a complex number slightly differently – instead of giving the x and y coordinates, we will give a distance r (the modulus) and angle \theta (the argument). We call this the polar form of a complex number.

Many amazing properties of complex numbers are revealed by looking at them in polar form! Let’s learn how to convert a complex number a+bi into polar form, and back again.

Definition 21.4. Let a+bi be a complex number. The absolute value of a+bi, denoted by |a+bi|, is the distance between the point a+bi in the complex plane and the origin (0,0). By the Pythagorean Theorem, we can calculate the absolute value of a+bi as follows:

    \[|a+bi|=\sqrt{a^2+b^2}\]

Definition 21.6. Let a+bi be a complex number. The coordinates in the plane can be expressed in terms of the absolute value, or modulus, r=|a+bi| and the angle, or argument, \theta formed with the positive real axis (the x-axis) as shown in the diagram:

Expressing a complex number in terms of absolute value and angle.
Expressing a complex number a+bi in terms of absolute value and angle.

As shown in the diagram, the coordinates a and b are given by:
a=r\cdot\cos(\theta), \text{ and } b=r\cdot\sin(\theta)

Substituting and factoring out r, we can use these to express a+bi in polar form:

Polar form: a+bi = r\left(\cos(\theta) + i\cdot\sin(\theta)\right)

How do we find the modulus r and the argument \theta?

Note that r is given by the absolute value. For \theta, we note that \frac{b}{a}=\frac{r \cdot \sin (\theta)}{r \cdot \cos (\theta)}=\frac{\sin (\theta)}{\cos (\theta)}=\tan (\theta). This leads to the following:

Formulas for converting to polar form (finding the modulus r and argument \theta): r=\sqrt{a^2+b^2}, \tan(\theta)=\frac{b}{a}

With regards to the modulus \theta, we can certainly use the inverse tangent function \arctan\left(\frac{b}{a}\right). However, we have to be a little careful: since the arctangent only gives angles in Quadrants I and II, we need to doublecheck the quadrant of (a,b).

If \arctan\left(\frac{b}{a}\right) is in the correct quadrant then \theta=\arctan\left(\frac{b}{a}\right). If not, then we add \pi radians or 180^\circ to obtain the angle in the opposing quadrant: \theta=\arctan\left(\frac{b}{a}\right)+\pi, or \theta=\arctan\left(\frac{b}{a}\right)+180^\circ. You’ll see this in action in the following example.

Example 21.7. Convert the complex number to polar form.
a) 2+3 i
b) -2-2 \sqrt{3} i
c) 4-3 i
d) -4 i

VIDEO: Converting complex numbers to polar form – Example 21.7

Example 21.8. Convert the number from polar form into the standard form a+bi
a) 3 \cdot\left(\cos \left(117^{\circ}\right)+i \sin \left(117^{\circ}\right)\right)
b) 4 \cdot\left(\cos \left(\frac{5 \pi}{4}\right)+i \sin \left(\frac{5 \pi}{4}\right)\right)

VIDEO: Converting complex numbers from polar form into standard form – Example 21.8

Multiplication and division of complex numbers in polar form

Why is polar form useful? The primary reason is that it gives us a simple way to picture how multiplication and division work in the plane. The proposition below gives the formulas, which may look complicated – but the idea behind them is simple, and is captured in these two slogans:

When we multiply complex numbers: we multiply the rs and add the \thetas.
When we divide complex numbers: we divide the rs and subtract the \thetas

Proposition 21.9. Let r_{1}\left(\cos \left(\theta_{1}\right)+i \sin \left(\theta_{1}\right)\right) and r_{2}\left(\cos \left(\theta_{2}\right)+i \sin \left(\theta_{2}\right)\right) be two complex numbers in polar form. Then, the product and quotient of these are given by

r_{1}\left(\cos \left(\theta_{1}\right)+i \sin \left(\theta_{1}\right)\right) \cdot r_{2}\left(\cos \left(\theta_{2}\right)+i \sin \left(\theta_{2}\right)\right) \ =r_{1} r_{2} \cdot\left(\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right)

\frac{r_{1}\left(\cos \left(\theta_{1}\right)+i \sin \left(\theta_{1}\right)\right)}{r_{2}\left(\cos \left(\theta_{2}\right)+i \sin \left(\theta_{2}\right)\right)} =\frac{r_{1}}{r_{2}} \cdot\left(\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right)

Example 21.10. Multiply or divide the complex numbers, and write your answer in polar and standard form.
a) 5\left(\cos \left(11^{\circ}\right)+i \sin \left(11^{\circ}\right)\right) \cdot 8\left(\cos \left(34^{\circ}\right)+i \sin \left(34^{\circ}\right)\right)
b) \quad 3\left(\cos \left(\frac{5 \pi}{8}\right)+i \sin \left(\frac{5 \pi}{8}\right)\right) \cdot 12\left(\cos \left(\frac{7 \pi}{8}\right)+i \sin \left(\frac{7 \pi}{8}\right)\right)
c) \frac{32\left(\cos \left(\frac{\pi}{4}\right)+i \sin \left(\frac{\pi}{4}\right)\right)}{8\left(\cos \left(\frac{7 \pi}{12}\right)+i \sin \left(\frac{7 \pi}{12}\right)\right)}
d) \frac{4\left(\cos \left(203^{\circ}\right)+i \sin \left(203^{\circ}\right)\right)}{6\left(\cos \left(74^{\circ}\right)+i \sin \left(74^{\circ}\right)\right)}

e) INTUITIVE BONUS: Without doing any calculation or conversion, describe where in the complex plane to find the number obtained by multiplying (5+2i)(-1+6i).

VIDEO: Multiplication and division of complex numbers in polar form – Example 21.10

That’s it for today! Give the WeBWorK a try, and let me know if you have any questions.