Category: Lessons (Page 4 of 4)

Lesson 21: Complex numbers

April 28, 2020 / Jonas Reitz / 1 Comment

Hi everyone! Read through the material below, watch the videos.

Lesson 21: Complex Numbers

Topic: This lesson covers Chapter 21: Complex numbers.
WeBWorK: There are four WeBWorK assignments on today’s material:
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.

Lesson 21_ Complex numbers Download

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.

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 $r$s and add the $\theta$s.
When we divide complex numbers: we divide the $r$s and subtract the $\theta$s

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.

Lesson 20: Trigonometric Equations

April 23, 2020 / Jonas Reitz / 0 Comments

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 20: Trigonometric Equations

Lesson Date: Thursday, April 23rd.
Topic: This lesson covers Chapter 20: Trigonometric Equations.
WeBWorK: There is one WeBWorK assignment on today’s material, due next Thursday 4/30:
Trigonometry – Equations

Question of the Day: If we know $\sin(x)=\frac{1}{2}$, what do we know about the angle $x$?

Lesson NOtes (Notability – pdf):

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

Lesson 20_ Trigonometric Equations Download

Basic Trigonometric Equations

Equations of the form tan(x)=c

Example 20.1. Solve for $x$: $\tan(x)=\sqrt{3}$

VIDEO: Example 20.1, tan(x)=c

Observation 20.2. To solve $\tan(x)=c$, we first determine one solution $x=\tan^{-1}(c)$. Then the general solution is given by:
$$x=\tan^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

EQUATIONS OF THE FORM cos(x)=c

Example 20.4. Solve for $x$: $\cos(x)=\frac{1}{2}$

VIDEO: Example 20.4, cos(x)=c

Observation 20.5. To solve $\cos(x)=c$, we first determine one solution $x=\cos^{-1}(c)$. Then the general solution is given by:
$$x=\pm\cos^{-1}(c)+2n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

EQUATIONS OF THE FORM sin(x)=c

Example 20.7. Solve for $x$: $\sin(x)=\frac{\sqrt{2}}{2}$

VIDEO: Example 20.7, sin(x)=c

Observation 20.8. To solve $\sin(x)=c$, we first determine one solution $x=\sin^{-1}(c)$. Then the general solution is given by:
$$x=(-1)^{n}\cdot\sin^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

Summary: We summarize the different formulas used to solve the basic trigonometric equations in the following table.

Solve: $\sin(x)=c$	Solve: $\cos(x)=c$	Solve: $\tan(x)=c$
First, find one solution, that is: $\sin^{-1}(c)$. Use: $\sin^{-1}(-c)=-\sin^{-1}(c)$	First, find one solution, that is: $\cos^{-1}(c)$. Use: $\cos^{-1}(-c)=\pi-\cos^{-1}(c)$	First, find one solution, that is: $\tan^{-1}(c)$. Use: $\tan^{-1}(-c)=-\tan^{-1}(c)$
The general solution is: $x=(-1)^{n} \sin^{-1}(c)+n \pi$	The general solution is: $x=\pm \cos^{-1}(c)+2 n \pi$	The general solution is: $x=\tan ^{-1}(c)+n \pi$
where $n=0,\pm 1,\pm 2, \ldots$	where $n=0,\pm 1,\pm 2, \ldots$	where $n=0,\pm 1,\pm 2, \ldots$

Example 20.10. Find the general solution of the equation, and state at least $5$ distinct solutions.
a) $\sin(x)=-\frac{1}{2}$
b) $\cos(x)=-\frac{\sqrt{3}}{2}$

Equations involving trigonometric functions

Example 20.11. Solve for $x$
a) $2 \sin (x)-1=0$
b) $\sec (x)=-\sqrt{2}$
c) $7 \cot (x)+3=0$

VIDEO: Example 20.11 – equations with trig functions (linear)

Example 20.12. Solve for $x$.
a) $\tan ^{2}(x)+2 \tan (x)+1=0 \quad$ b) $2 \cos ^{2}(x)-1=0$

VIDEO: Example 20.12 – equations with trig functions (quadratic)

Lesson 19: Inverse trigonometric functions

April 21, 2020 / Jonas Reitz / 1 Comment

Hi everyone! Read through the material below, watch the videos.

Lesson 19: Inverse trigonometric functions

Topic: This lesson covers Chapter 19: Inverse trigonometric functions.
WeBWorK: There is one WeBWorK assignment on today’s material:
Trigonometry – Inverse Functions

Question of the Day: Are the trigonometric functions $\sin(x),\cos(x)$ and $\tan(x)$ one-to-one functions?

Lesson Notes (Notability – pdf):

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

Lesson 19_ Inverse trigonometric functions Download

The functions $\sin^{-1}, \cos^{-1},\tan^{-1}$

In this section, we are interested in the inverse functions of the trigonometric functions $y=\sin(x), y=\cos(x),$ and $y=\tan(x)$. You may recall from our work earlier in the semester that in order for a function to have an inverse, it must be one-to-one (or pass the horizontal line test: any horizontal line intersects the graph at most once).

The function $\tan^{-1}(x)$

Recall the graph of the function $y=\tan(x)$:

Graph of y=tan(x) — Graph of $y=\tan(x)$.

Notice that since the graph consists of a repeating pattern of vertical stripes, any horizontal line will touch the graph in multiple places – this graph FAILS the horizontal line test (it is NOT one-to-one). How can we define the inverse? By restricting the domain – that is, only looking at one of the repeating vertical stripes. If we only look at the part of the graph between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ then the function is one-to-one (that it, the red part of the function above is, by itself, one-to-one).

Definition 19.1. The inverse of the function $y=\tan (x)$ with restricted domain $D=\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ and range $R=\mathbb{R}$ is called the inverse tangent or arctangent function. It is denoted by:
$y=\tan ^{-1}(x) \quad$ or $\quad y=\arctan (x) \quad \Longleftrightarrow \quad \tan (y)=x, \quad y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Note that the inverse tangent function is written both $\tan^{-1}(x)$ and $\arctan(x)$ — they mean the same thing.

Observation: The inverse tangent is an odd function, so $\tan^{-1}(-x)=-\tan^{-1}(x)$
(recall that a function $f(x)$ is odd provided $f(-x)=-f(x)$)

Example 19.1 Recall the exact values of the tangent function from Chapter 17:

Use the table and Observation above to find exact values of the inverse tangent function. Give answers in both degrees and radians.
a. $\arctan(1)$ b. $\arctan\left(-\frac{\sqrt{3}}{3}\right)$. c. $\tan^{-1}(0)$

VIDEO: The Inverse Tangent Function – Definition and Example 19.1

THE FUNCTION $\sin^{-1}(x)$

Consider the graph of the function $y=\sin(x)$. It is not one-to-one either:

The graph of sin(x). — The graph of $\sin(x)$.

However, if we restrict the function to the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ (shown in red) the resulting function is one-to-one, and so we can consider the inverse function.

Definition 19.5. The inverse of the function $y=\sin (x)$ with restricted domain $D=\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ and range $R=[-1,1]$ is called the inverse sine or arcsine function. It is denoted by
$y=\sin ^{-1}(x) \quad \text { or } \quad y=\arcsin (x) \quad \Longleftrightarrow \quad \sin (y)=x, \quad y \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
The arcsine reverses the input and output of the sine function, so that the arcsine has domain $D=[-1,1]$ and range $R=\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$.

Graph of the inverse sine, or arcsine, function y=arcsin(x). — Graph of the inverse sine, or arcsine, function $y=\sin^{-1}(x)$

Observation: The inverse sine function is an odd function, so $\sin^{-1}(-x)=-\sin^{-1}(x)$.

Example 19.7. Recall the values of the sine function for common angles:

Values of sin(x) for common angles. — Values of $\sin(x)$ for common angles.

Use the Table and Observation above to find exact values of the arcsine function. Give answers in both degrees and radians.
a. $\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)$, b. $\sin^{-1}(1)$, c. $\sin^{-1}(0)$, d. $\sin ^{-1}\left(\frac{-1}{2}\right)$, e. $\sin^{-1}(3)$

VIDEO: The Inverse Sine Function – Definition and Example 19.7

THE FUNCTION $\cos^{-1}(x)$

We treat the function $\cos(x)$ similar to $\sin(x)$. However, we are no longer able to use the interval $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$. Why?

The graph of cos(x) — The graph of $\cos(x)$

In order to make the cosine function one-to-one, we restrict to the interval $[0,\pi]$.

Definition 19.8. The inverse of the function $y=\cos (x)$ with restricted domain $D=[0, \pi]$ and range $R=[-1,1]$ is called the inverse cosine or arccosine function. It is denoted by
$y=\cos ^{-1}(x) \quad \text { or } \quad y=\arccos (x) \quad \Longleftrightarrow \quad \cos (y)=x, \quad y \in[0, \pi]$
The arccosine reverses the input and output of the cosine function, so that the arccosine has domain $D=[-1,1]$ and range $R=[0, \pi]$.

The graph of arccos(x). — The graph of $\arccos(x)$.

Observation: The arccosine function is neither even nor odd. However, it does obey the following symmetry: $\cos^{-1}(-x)=\pi-\cos^{-1}(x)$
(in many problems, you can avoid the use of this formula by remembering the unit circle definition of cosine).

Example 19.10. Recall the values of the cosine function for common angles:

Values of cos(x) for common angles — Values of $\cos(x)$ for common angles.

Use the Table and Observation above to find exact values of the arccosine function. Give answers in both degrees and radians.
a. $\arccos\left(\frac{\sqrt{3}}{2}\right)$, b. $\cos^{-1}(1)$, c. $\cos ^{-1}(0)$, d. $\arccos=\left(-\frac{1}{2}\right)$, e. $\arccos(2)$

VIDEO: The Inverse Cosine Function – Definition and Example 19.10

Inverse trig functions on the TI-84+ calculator

How do we find values of inverse trig functions that don’t appear in our “common angles” table?

Example. Find the values of the inverse trig functions using a calculator. Include at least 5 decimal digits past the decimal point.

a. $\arccos(0.35)$ (in radians)
b. $\tan^{-1}(-13.2)$ (in degrees)

VIDEO: Inverse trig functions on the calculator

RECALL: Converting between radians and degrees

$\text{radians}=\text{degrees}\cdot\frac{\pi}{180}$
$\text{degrees}=\text{radians}\cdot\frac{180}{\pi}$

Lesson 18: Addition of angles and multiple angle formulas

April 16, 2020 / Jonas Reitz / 2 Comments

Hi everyone! Read through the material below, watch the videos, collect your questions.

Lesson 18: Addition of angles and multiple angle formulas

Topic: This lesson covers Chapter 18: Addition of angles and multiple angle formulas.
WeBWorK: There are two WeBWorK assignments on today’s material:
Trigonometry – Sum and Difference Formulas
Trigonometry – Double and Half Angle Formulas

Lesson Notes (Notability – pdf):

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

Lesson 18_ Addition of angles and multiple angle formulas Download

Question of the Day: Suppose we know the values of the trig functions of two angles $a$ and $b$. Can we use them to find the values of the trig functions of the angle $a+b$?

Today, we will see how we can do exactly that – the idea is to work with formulas that allow us to calculate, for example, $\sin(a+b)$ and $\cos(a+b)$ based on the values of $\sin(a),\cos(a),\sin(b),$ and $\cos(b)$.

NOTE: We often use greek letters for angles — this helps us keep track of what’s an angle and what’s not. The most common are the greek letters alpha $\alpha$ and beta $\beta$. We’ll be using these instead of $a$ and $b$.

Warning: the videos for today’s lecture are *quite long* – however, they consist almost entirely of examples, with a lot of explanation. Feel free to skip around, or to try the WeBWorK first (if you get stuck, the videos might help).

Addition and Subtraction of Angles

Proposition 18.1. For any angles $\alpha$ and $\beta$,

$\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$
$\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta$
$\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$
$\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$
$\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}$
$\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}$

Where did these formulas come from!?

Great question! To answer it, you need to see the *proof* of these formulas – this appears in your book in Chapter 18.

Now, we are going to see how these formulas let us calculate the values of trig functions at many different angles, based on just a few common angles (such as those listed in the table below – if you don’t know them, this is a great time to learn them!).

Values of trig function for common angles

Example 18.2. Find the exact values of the trigonometric functions:

a) $\cos \left(\frac{\pi}{12}\right)$
b) $\tan \left(\frac{5 \pi}{12}\right)$
c) $\cos \left(\frac{11 \pi}{12}\right)$

VIDEO: Example 18.2 applying angle sum and difference formulas

Double and Half Angles

Proposition 18.5. Let $\alpha$ be any angle. Then we have the half-angle formulas:

$\sin \frac{\alpha}{2} =\pm \sqrt{\frac{1-\cos \alpha}{2}}$
$\cos \frac{\alpha}{2} =\pm \sqrt{\frac{1+\cos \alpha}{2}}$
$\tan \frac{\alpha}{2} =\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}=\pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}$

and the double-angle formulas:

$\sin (2 \alpha) =2 \sin \alpha \cos \alpha$
$\cos (2 \alpha) =\cos ^{2} \alpha-\sin ^{2} \alpha=1-2 \sin ^{2} \alpha=2 \cos ^{2} \alpha-1$
$\tan (2 \alpha) &=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}$

Example 18.6. Find the exact values of the trigonometric functions:

a) $\sin \left(\frac{\pi}{8}\right)$
b) $\cos \left(\frac{9 \pi}{8}\right)$
c) $\tan \left(\frac{\pi}{24}\right)$

VIDEO: Example 18.6 applying half-angle formulas

Example 18.7. Find the trigonometric functions of $2\alpha$ when $\alpha$ has the properties below.

a) $\sin (\alpha)=\frac{3}{5},$ and $\alpha$ is in quadrant 1
b) $\tan (\alpha)=\frac{12}{5},$ and $\alpha$ is in quadrant 2

VIDEO: Example 18.7 applying double-angle formulas

That’s it for now. Take a look at the WeBWorK!

Lesson 17: Trigonometric functions

April 14, 2020 / Jonas Reitz / 2 Comments

Hi everyone! Read through the material below, watch the videos, and collect your questions.

Lesson 17: Trigonometric functions

Topic: This lesson covers Chapter 17: Trigonometric functions.
WeBWorK: There are five WeBWorK assignments on today’s material:
Trigonometry - Unit Circle,
Trigonometry - Graphing Amplitude,
Trigonometry - Graphing Period,
Trigonometry - Graphing Phase Shift, and
Trigonometry - Graphing Comprehensive

Today we start trigonometric functions. We’ll begin with a review of the basics of trigonometry — if you remember everything about trigonometry, you can skip this part (but please don’t!). Then we’ll think about how these behave as functions, and look at their graphs.

Lesson Notes (Notability – pdf):

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

Lesson 17_ Trigonometric Functions Download

Review of trigonometry

a. Angles

We measure angles using two different systems, degrees and radians. In degrees, a full circle is $360^\circ$. In radians, a full circle is $2\pi \text{rad}$. Thus, we have $360^\circ=2\pi$ radians. If we consider a half-circle (divide both sides by two), we get:

$$180^\circ=\pi$$

Although most people are more familiar with degrees from their day-to-day lives, most mathematics at this level and above use radians. It will help you to get familiar with common angles in radians! The first video gives a review.

Why do mathematicians and scientists prefer radians to degrees? Great question!

VIDEO: Review of angles, radian measure

b. Trigonometric Functions

Definition. To define the trigonometric functions, we consider the following diagram: the initial side of an angle $x$ lies on the positive x-axis, and the terminal side of the angle passes through a point $P(a,b)$.

Diagram: Definition of Trigonometric Functions

The distance from the origin $(0,0)$ to the point $P$ is $r$ (note: if we know $a$ and $bg$ we can calculate $r$ by using the Pythagorean Theorem, $r=\sqrt{a^2+b^2}$). Then the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) of $x$ are defined:

$$\sin(x)=\frac{b}{r},\quad \cos(x)=\frac{a}{r},\quad\tan(x)=\frac{a}{b}$$
$$\csc(x)=\frac{r}{b},\quad \sec(x)=\frac{r}{a},\quad\cot(x)=\frac{b}{a}$$

VIDEO: Review – definitions of trigonometric functions

Next, we need to know the values of the trig functions for some common angles. Once again, this is a review of material from previous courses – the following video will take you through some examples, but for more details I recommend checking out the videos on our video resource page.

VIDEO: Values of trig functions at common angles

Special Triangles: These allow us to find the values of the trig functions at various common angles.

These are the values of sine, cosine and tangent for common angles in the first quadrant. Learn them, memorize them, get to know them!

This is the end of the ‘review’ part of the lesson. Need a little more help? Take a look at the videos on our video resource page.

Graphs of trigonometric functions

Now we turn to the main idea of this lesson. We begin by looking at the graphs of the basic trig functions, $\sin x$, $\cos x$, and $\tan x$.

VIDEO: Graphs of basic trig functions

Example 17.8. Graph the following functions:
$f(x)=\sin (x)+3,\quad g(x)=4 \cdot \sin (x), \quad h(x)=\sin (x+2),\quad i(x)=\sin (3 x)$
$j(x)=2 \cdot \cos (x)+3, \quad k(x)=\cos (2 x-\pi), \quad l(x)=\tan (x+2)+3$

VIDEO: Example 17.8 modifying the graphs of trig functions – amplitude, period, phase shift.

Definition 17.9. Let $f$ be one of the functions:
$f(x)=a \cdot \sin (b \cdot x+c) \quad \text { or } \quad f(x)=a \cdot \cos (b \cdot x+c)$
The number $|a|$ is called the amplitude, the number $\left|\frac{2 \pi}{b}\right|$ is the period, and the number $\frac{-c}{b}$ is called the phase shift.

That’s it for now. Take a look at the WeBWorK!

Lesson 15-16: Applications of Exponential and Log Functions

April 2, 2020 / Jonas Reitz / 1 Comment

Hi everyone! Read through the material below, watch the videos, collect your questions.

Lesson 15-16: Applications of Exponential and Logarithmic Functions

Topic: This lesson covers Chapter 15: Applications of Exponential and Logarithmic Functions and Chapter 16: Half-Life and Compound Interest.
WeBWorK: There is one WeBWorK assignment on today’s material: Exponential Functions - Growth and Decay.

We study exponential functions because they are amazingly good at describing real-world phenomena. Today we will look at several different applications of exponential functions, including population growth and virus spread.

Lesson Notes (Notability – pdf):

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

Lesson 15-16_ Applications of Exponential and Log Functions Download

Exponential growth and decay

RECALL: If $f(x)$ is an exponential function, then $f(x)=c\cdot b^x$.

NOTE: You can determine the values of $c$ and $b$ from any two points (input-output) on the graph.

Example 15.1.
a) $f(0)=4, \quad f(1)=20$
c) $f(2)=160, \quad f(7)=5$

Example 15.3 The population size of a country was $12.7$ million in the year 2000, and $14.3$ million in the year 2010.
a) Assuming an exponential growth for the population size, ﬁnd the formula for the population depending on the year $t$ (where $t=0$ in the year 2000.
b) What will the population size be in the year 2015, assuming the formula holds until then?
c) When will the population reach 18 million?

You will often see exponential growth and decay functions written in a slightly different (but equivalent) form, using the number $e = 2.718…$ as a base. It’s also traditional to use $t$ (for time) instead of $x$ as our independent variable.

FACT. If $f(t)$ is an exponential function, then $f$ can be written as $f(t)=Pe^{rt}$. In this form:

$t$ represents time
$P$ is the initial amount
$r$ is the growth rate (if $r$ is positive, we have exponential growth, if $r$ is negative we have exponential decay)
$f(t)$ is the amount remaining at time t
$e=2.718…$

QUESTION: What’s the connection between $f(x)=c\cdot b^x$ and $f(t)=Pe^{rt}$?

Definition (exponential decay). The half-life of a substance is the time it takes for the amount to be cut in half.

EXAMPLE: A study published on March 17th, in the New England Journal of Medicine found experimentally that the half-life of the Covid-19 virus in the air is approximately 1.15 hours. A single cough by an infected person can release up to 6 billion coronavirus molecules into the air. Let’s consider what happens after a single cough by an infected person.

a. Model the number of remaining virus molecules $V(t)$ in the air at time $t$ by an exponential function $V(t)=Pe^{rt}$ (find $P,r$).
b. How many of virus molecules will remain viable 5 hours after the person coughed?
c. How long will it take for the number of remaining molecules to reach $6$ million ($0.1\%$ of the original amount)?

That’s it for now. Take a look at the WeBWorK!

Lesson 14: Properties of Exponential and Logarithmic Functions

March 26, 2020 / Jonas Reitz / 4 Comments

Hi everyone! Read through the material below, watch the videos, and send your professor your questions.

Lesson 14: Properties of Exponential and Logarithmic Functions

Topic: This lesson covers Chapter 14 in the book, Exponential and Logarithmic Functions.
WeBWorK: There are three WeBWorK assignments on today’s material: Logarithmic Functions - Properties, Logarithmic Functions - Equations, and Exponential Functions - Equations

Today’s goal is to review the properties/rules of exponents and logs, and then see how we can use them to solve equations.

Lesson Notes (Notability – pdf):

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

Lesson 14_ Properties of Exponential and Logarithmic Functions Download

Properties of Exponential and Logarithmic Functions

You should already be familiar with the rules of exponents and rules of logarithms. I realize you may not remember them, so read through the material below and take a look at the examples in the first video.

Review: Properties of Exponential Functions

The following rules apply to exponential functions (where $b>0$ and $x,y$ are any real numbers):

$b^{x+y}=b^{x} \cdot b^{y}$
$b^{x-y}=\frac{b^{x}}{b^{y}}$
$\left(b^{x}\right)^{n}=b^{n x}$

Review: Properties of Logarithmic Functions

The following rules apply to logarithmic functions (where $b>0, b\neq 1$ and $x,y > 0$, and $n$ is an integer).

$\log_{b}(x \cdot y)=\log_{b}(x)+\log_{b}(y)$
$\log_{b}\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)$
$\log_{b}\left(x^{n}\right)=n \cdot \log_{b}(x)$ 

Change of base formula (if $a>0, a\neq 1)$:  
$\log_{b}(x)=\frac{\log_{a}(x)}{\log_{a}(b)}$

Since the logarithm is the inverse of the exponential function, each rule of exponents has a corresponding rule of logarithms.

Example 14.1: Combine the terms using the properties of logarithms so as to write as one logarithm.

a) $\frac{1}{2} \ln (x)+\ln (y)$
b) $\frac{2}{3}\left(\log \left(x^{2} y\right)-\log \left(x y^{2}\right)\right)$
c) $2 \ln (x)-\frac{1}{3} \ln (y)-\frac{7}{5} \ln (z)$ 
d) $5+\log_{2}\left(a^{2}-b^{2}\right)-\log_{2}(a+b)$

Example 14.2:  Write the expressions in terms of elementary logarithms $u=$ $\log_{b}(x), v=\log_{b}(y),$ and, in part $(\mathrm{c}),$ also $w=\log_{b}(z) .$ Assume that $x, y, z>0$

a) $\ln \left(\sqrt{x^{5}} \cdot y^{2}\right)$
b) $\log (\sqrt{\sqrt{x} \cdot y^{3}}) \quad$ 
c) $\log _{2}(\sqrt[3]{\frac{x^{2}}{y \sqrt{z}}})$

Warning: The videos in this lesson are LONG – about 30 minutes each – but they consist almost entirely of EXAMPLES. Feel free to skip around.

Solving Exponential and Log Equations

Now we’re going to use these properties to solve equations.

Example 14.5: Solve for $x$.

a) $2^{x+7}=32$
b) $10^{2 x-8}=0.01$
c) $7^{2 x-3}=7^{5 x+4}$
d) $5^{3 x+1}=25^{4 x-7}$
e) $\ln (3 x-5)=\ln (x-1)$
f) $\log_{2}(x+5)=\log_{2}(x+3)+4$
g) $\log_{6}(x)+\log_{6}(x+4)=\log_{6}(5)$ 
h) $\log_{3}(x-2)+\log_{3}(x+6)=2$

Example 14.6: Solving Log Equations

a) $3^{x+5}=8$
b) $13^{2 x-4}=6$
c) $5^{x-7}=2^{x}$
d) $5.1^{x}=2.7^{2 x+6}$
e) $17^{x-2}=3^{x+4}$
f) $7^{2x+3}=11^{3x-6}$

That’s it for today, everybody! Give the WeBWorK a try.

Lesson 13: Exponential and Logarithmic Functions

March 24, 2020 / Jonas Reitz / 6 Comments

Hi everyone! Read through the material below, watch the videos, and send me your questions.

Lesson 13: Exponential and Logarithmic Functions

Topic: This lesson covers Chapter 13 in the book, Exponential and Logarithmic Functions.
WeBWorK: There are two WeBWorK assignments on today’s material: Exponential Functions - Graphs and Logarithmic Functions - Graphs .

Lesson Notes (Notability – pdf):

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

Lesson 13_ Exponential and Logarithmic Functions Download

Exponential Functions and their Graphs

We’ve been living in the world of Polynomials and Rational Functions. We now turn to exponential functions. These functions are “very natural” – that is, they show up in the real world – but they are also more complicated than Polynomial and Rational functions (for example, an exponential function grows more quickly than any Polynomial)

The spread of coronavirus, like other infectious diseases, can be modeled by exponential functions.

Definition. An exponential function is a function of the form $f(x)=c\cdot b^x$, where $b$ and $c$ are real numbers and $b$ is positive ($b$ is called the base, $x$ is the exponent).

Example 1 (Textbook 13.2): Graph the exponential functions $f(x)=2^x, g(x)=3^x, h(x)=10^x, k(x)=\left(\frac{1}{2}\right)^x, l(x)=\left(\frac{1}{10}\right)^x$.

Now let’s see what happens when we change the number $c$ in $y=c\cdot b^x$.

Example 2 (Textbook 13.6): Graph the exponential functions
a) $y=2^{x}, \quad$ b) $y=3 \cdot 2^{x}, \quad$ c) $y=(-3) \cdot 2^{x}, \quad$ d) $y=0.2 \cdot 2^{x}, \quad$ e) $y=(-0.2) \cdot 2^{x}$

Example 3: The graph below shows an exponential function $f(x)$. Find a formula for $f(x)$.

Logarithmic Functions and their Graphs

Definition. If $b$ is a positive real number and $b\neq 1$, then the logarithm with base $b$ is defined:
$y=\log_b(x) \iff b^y=x$

What does the definition of logarithm mean? The idea is that the logarithm is the inverse function of the exponential function. Let’s look at an example.

Question: Is an exponential function one-to-one? (What does one-to-one means).

Example 4. The graph below shows the function $y=\log_2(x)$ but shifted to the right 3 units. Find a formula for the function in the graph.

That’s it for now! Take a look at the WeBWorK assignment, leave your questions below (or use the Ask for Help button in WeBWorK, or send me an email)

Lesson 12 Followup – the Magic

March 21, 2020 / Jonas Reitz / 0 Comments

Hi everyone,

In one of the examples during Lesson 12 last week (it was Example 2), there was a place where we needed to solve the following equation:

$x^4-5x^3-x^2+5x=0$

In the lesson, I used magic to find the roots — in this video, I’ll actually go through the steps:

Lesson 12: Polynomial and Rational Inequalities

March 19, 2020 / Jonas Reitz / 4 Comments

Hi everyone! Read through the material below, watch the videos, and send me your questions.

Lesson 12: Polynomial and Rational Inequalities

Topic: This lesson covers Chapter 12 in the book, Polynomial and Rational Inequalities.
WeBWorK: There are two WeBWorK assignments on today’s material: Polynomials - Inequalities, and Rational Functions - Inequalities.

Introduction to polynomial inequalities

Definition. A polynomial inequality is an inequality (which means it uses one of these: $<,\leq,>,\geq$ instead of an equals sign) with a polynomial on each side

Example 1: $x^2-3x-4\geq 0$

We’re interested in solving these inequalities, which means answering the question: “For which real numbers x is the inequality true?”

Now let’s look at the same example, and see how to solve it without looking at the graph:

Example 2: Solve $x^{4}-x^{2}>5\left(x^{3}-x\right)$

Example 2, concluded:

Rational inequalities

What happens if we allow rational functions instead of just polynomials?

Example 3: Solve $\frac{x^{2}-5 x+6}{x^{2}-5 x} \geq 0$

Good job! You are now ready to practice on your own. Take a look at the WeBWorK assignment, and don’t forget to use the “Ask for Help” button if you get stuck.

Here are more video resources if you’d like to see additional examples.

ASSIGNMENT: Watch videos, try webwork.

Lesson Notes (Notability – pdf):

Review of Complex Numbers

Polar form

Multiplication and division of complex numbers in polar form

Lesson NOtes (Notability – pdf):

Basic Trigonometric Equations

Equations of the form tan(x)=c

EQUATIONS OF THE FORM cos(x)=c

EQUATIONS OF THE FORM sin(x)=c

Equations involving trigonometric functions

Lesson Notes (Notability – pdf):

The functions $\sin^{-1}, \cos^{-1},\tan^{-1}$

The function $\tan^{-1}(x)$

THE FUNCTION $\sin^{-1}(x)$

THE FUNCTION $\cos^{-1}(x)$

Inverse trig functions on the TI-84+ calculator

RECALL: Converting between radians and degrees

Lesson Notes (Notability – pdf):

Addition and Subtraction of Angles

Where did these formulas come from!?

Double and Half Angles

Lesson Notes (Notability – pdf):

Review of trigonometry

a. Angles

b. Trigonometric Functions

Graphs of trigonometric functions

Lesson Notes (Notability – pdf):

Exponential growth and decay

Lesson Notes (Notability – pdf):

Properties of Exponential and Logarithmic Functions

Review: Properties of Exponential Functions

Review: Properties of Logarithmic Functions

Solving Exponential and Log Equations

Lesson Notes (Notability – pdf):

Exponential Functions and their Graphs

Logarithmic Functions and their Graphs

Introduction to polynomial inequalities

Rational inequalities

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