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Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 17: Trigonometric functions
Topic. This lesson covers Session X: lessontopic
Learning Outcomes.
WeBWorK. There are X WeBWorK assignments on today’s material:
Lesson Notes. Coming Soon
Additional Video Resources.
Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 20: Trigonometric equations
Topic. This lesson covers Session X: lessontopic
Learning Outcomes.
WeBWorK. There are X WeBWorK assignments on today’s material:
Lesson Notes. Coming Soon
Additional Video Resources.
Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 19: Inverse trigonometric functions
Topic. This lesson covers Session X: lessontopic
Learning Outcomes.
WeBWorK. There are X WeBWorK assignments on today’s material:
Lesson Notes. Coming Soon
Additional Video Resources.
Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 18: Addition of angles and multiple angle formulas
Topic. This lesson covers Session X: lessontopic
Learning Outcomes.
WeBWorK. There are X WeBWorK assignments on today’s material:
Lesson Notes. Coming Soon
Additional Video Resources.
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 $x$?
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
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:
$$x=\tan^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$
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:
$$x=\pm\cos^{-1}(c)+2n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$
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) $\cos(x)=-\frac{\sqrt{3}}{2}$
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)
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
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
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).
Proposition 18.1. For any angles $\alpha$ and $\beta$,
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!).
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
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!
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, andTrigonometry - 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.
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
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
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 $P(a,b)$.
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
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.
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!
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