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.

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)

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 $y=\tan(x)$:

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:

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 $R=\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$.

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:

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?

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 $R=[0, \pi]$.

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:

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)

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 $\beta$,

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

Double and Half Angles

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

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.

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.

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 $P(a,b)$.

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:

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.

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.

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