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Algebra 2 and Trigonometry

Ann Xavier Gantert

Chapter 13

TRIGONOMETRIC EQUATIONS

Educators


Problem 1

Explain why the solution set of the equation $2 x+4=8$ is $\{2\}$ but the solution set of the equation $2 \sin x+4=8$ is $\{ \},$ the empty set.

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Problem 2

Explain why $2 x+4=8$ has only one solution in the set of real numbers but the equation $2 \tan x+4=8$ has infinitely many solutions in the set of real numbers.

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Problem 3

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
2 \cos \theta-1=0
$$

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Problem 4

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
3 \tan \theta+\sqrt{3}=0
$$

Joseph F.
Numerade Educator

Problem 5

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
4 \sin \theta-1=2 \sin \theta+1
$$

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Problem 6

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
5(\cos \theta+1)=5
$$

Joseph F.
Numerade Educator

Problem 7

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
3(\tan \theta-2)=2 \tan \theta-7
$$

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Problem 8

In $3-8,$ find the exact solution set of each equation if $0^{\circ} \leq \theta<360^{\circ} .$
$$
\sec \theta+\sqrt{2}=2 \sqrt{2}
$$

Joseph F.
Numerade Educator

Problem 9

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
3 \sin \theta-\sqrt{3}=\sin \theta
$$

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Problem 10

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
5 \cos \theta+3=3 \cos \theta+5
$$

Joseph F.
Numerade Educator

Problem 11

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
\tan \theta+12=2 \tan \theta+11
$$

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Problem 12

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
\sin \theta+\sqrt{2}=\frac{\sqrt{2}}{2}
$$

Joseph F.
Numerade Educator

Problem 13

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
3 \csc \theta+5=\csc \theta+9
$$

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Problem 14

In $9-14,$ find the exact values for $\theta$ in the interval $0 \leq \theta<2 \pi$
$$
4(\cot \theta+1)=2(\cot \theta+2)
$$

Joseph F.
Numerade Educator

Problem 15

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
\sin \theta+3=5 \sin \theta
$$

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Problem 16

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
3 \tan \theta-1=\tan \theta+9
$$

Joseph F.
Numerade Educator

Problem 17

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
5 \cos \theta+1=8 \cos \theta
$$

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Problem 18

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
4(\sin \theta+1)=6-\sin \theta
$$

Joseph F.
Numerade Educator

Problem 19

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
\csc \theta-1=3 \csc \theta-11
$$

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Problem 20

In $15-20,$ find, to the nearest degree, the measure of an acute angle for which the given equation is true.
$$
\cot \theta+8=3 \cot \theta+2
$$

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Problem 21

In $21-24,$ find, to the nearest tenth, the degree measures of all $\theta$ in the interval $0^{\circ} \leq \theta<360^{\circ}$ that make the equation true.
$$
8 \cos \theta=3-4 \cos \theta
$$

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Problem 22

In $21-24,$ find, to the nearest tenth, the degree measures of all $\theta$ in the interval $0^{\circ} \leq \theta<360^{\circ}$ that make the equation true.
$$
5 \sin \theta-1=1-2 \sin \theta
$$

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Problem 23

In $21-24,$ find, to the nearest tenth, the degree measures of all $\theta$ in the interval $0^{\circ} \leq \theta<360^{\circ}$ that make the equation true.
$$
\tan \theta-4=3 \tan \theta+4
$$

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Problem 24

In $21-24,$ find, to the nearest tenth, the degree measures of all $\theta$ in the interval $0^{\circ} \leq \theta<360^{\circ}$ that make the equation true.
$$
2-\sec \theta=5+\sec \theta
$$

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Problem 25

In $25-28,$ find, to the nearest hundredth, the radian measures of all $\theta$ in the interval $0 \leq \theta<2 \pi$ that make the equation true.
$$
10 \sin \theta+1=3-2 \sin \theta
$$

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Problem 26

In $25-28,$ find, to the nearest hundredth, the radian measures of all $\theta$ in the interval $0 \leq \theta<2 \pi$ that make the equation true.
$$
9-2 \cos \theta=8-4 \cos \theta
$$

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Problem 27

In $25-28,$ find, to the nearest hundredth, the radian measures of all $\theta$ in the interval $0 \leq \theta<2 \pi$ that make the equation true.
$$
15 \tan \theta-7=5 \tan \theta-3
$$

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Problem 28

In $25-28,$ find, to the nearest hundredth, the radian measures of all $\theta$ in the interval $0 \leq \theta<2 \pi$ that make the equation true.
$$
\cot \theta-6=2 \cot \theta+2
$$

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Problem 29

The voltage $E$ (in volts) in an electrical circuit is given by the function
$$
E=20 \cos (\pi t)
$$
where $t$ is time in seconds.
a. Graph the voltage $E$ in the interval $0 \leq t \leq 2$ .
b. What is the voltage of the electrical circuit when $t=1 ?$
c. How many times does the voltage equal 12 volts in the first two seconds?
d. Find, to the nearest hundredth of a second, the times in the first two seconds when the voltage is equal to 12 volts.
(1) Let $\theta=\pi t .$ Solve the equation $20 \cos \theta=12$ in the interval $0 \leq \theta<2 \pi$
(2) Use the formula $\theta=\pi t$ and your answers to part $(1)$ to find $t$ when $0 \leq \theta<2 \pi$ and the voltage is equal to 12 volts.

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Problem 30

A water balloon leaves the air cannon at an angle of $\theta$ with the ground and an initial velocity of 40 feet per second. The water balloon lands 30 feet from the cannon. The distance $d$ traveled by the water balloon is given by the formula
$$
d=\frac{1}{32} v^{2} \sin 2 \theta
$$
where $v$ is the initial velocity.
a. Let $x=2 \theta .$ Solve the equation $30=\frac{1}{32}(40)^{2} \sin x$ to the nearest tenth of a degree.
b. Use the formula $x=2 \theta$ and your answer to part a to find the measure of the angle that the cannon makes with the ground.

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Problem 31

It is important to understand the underlying mathematics before using the calculator to solve trigonometric equations. For example, Adrian tried to use the intersect feature of his graphing calculator to find the solutions of the equation cot $\theta=\sin \left(\theta-\frac{\pi}{2}\right)$ in the interval $0 \leq \theta \leq \pi$ but got an error message. Follow the steps that Adrian used to solve the equation:
(1) Enter $Y_{1}=\frac{1}{\tan X}$ and $Y_{2}=\sin \left(X-\frac{\pi}{2}\right)$ into the $Y=$ menu.
(2) Use the following viewing window to graph the equations:
$$
X \min =0, \operatorname{Xmax}=\pi, X s c l=\frac{\pi}{6}, Y \min =-5, Y \max =5
$$
(3) The curves seem to intersect at $\left(\frac{\pi}{2}, 0\right) .$ Press 2nd CALC 5 ENTER ENTER to select both curves. When the calculator asks for a guess, move the cursor near the intersection point using the arrow keys and then press ENTER
a. Why does the calculator return an error message?
b. Is $\theta=\frac{\pi}{2}$ a solution to the equation? Explain.

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