Precalculus

John W. Coburn

Chapter 6

Trigonometric Identities, Inverses, and Equations - all with Video Answers

Educators

Section 1

Fundamental Identities and Families of Identities

Problem 1

Three fundamental ratio identities are $\tan \theta=\frac{?}{\cos \theta}, \tan \theta=\frac{?}{\csc \theta},$ and $\cot \theta=\frac{?}{\sin \theta}$.

Andrew Bassila

Andrew Bassila

Numerade Educator

Problem 2

When applying identities due to symmetry, $\sin (-x) \tan x=$ ____ and $\cos (-x) \cot x=$ ____ .

Andrew Bassila

Numerade Educator

Problem 3

To show an equation is not an identity, we must find at least ____ value(s) where both sides of the equation are defined, but which makes the equation ____.

Andrew Bassila

Numerade Educator

Problem 4

Using a calculator we find $\sec ^{2} 45^{\circ}=$ ____ and $3 \tan 45^{\circ}-1=$ ____ We also find $\sec ^{2} 225^{\circ}=$ _____ and $3 \tan 225^{\circ}-1=$ _____ Is the equation $\sec ^{2} \theta=3 \tan \theta-1$ an identity?

Andrew Bassila

Numerade Educator

Problem 5

Use the pattern $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to add the following terms, and comment on this process versus "finding a common denominator:"
$$\frac{\cos x}{\sin x}-\frac{\sin x}{\sec x}$$

Andrew Bassila

Numerade Educator

Problem 6

Name at least four algebraic skills that are used with the fundamental identities in order to rewrite a trigonometric expression. Use algebra to quickly rewrite $(\sin x+\cos x)^{2}$.

Andrew Bassila

Numerade Educator

Problem 7

Starting with the ratio identity given, use substitution and fundamental identities to write four new identities belonging to the ratio family. Answers may vary.
$$\tan x=\frac{\sin x}{\cos x}$$

Andrew Bassila

Numerade Educator

Problem 8

Starting with the ratio identity given, use substitution and fundamental identities to write four new identities belonging to the ratio family. Answers may vary.
$$\cot x=\frac{\cos x}{\sin x}$$

Andrew Bassila

Numerade Educator

Problem 9

Starting with the Pythagorean identity given, use algebra to write four additional identities belonging to the Pythagorean family. Answers may vary.
$$1+\tan ^{2} x=\sec ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 10

Starting with the Pythagorean identity given, use algebra to write four additional identities belonging to the Pythagorean family. Answers may vary.
$$1+\cot ^{2} x=\csc ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 11

Verify the equation is an identity using multiplication and fundamental identities.
$$\sin x \cot x=\cos x$$

Andrew Bassila

Numerade Educator

Problem 12

Verify the equation is an identity using multiplication and fundamental identities.
$$\cos x \tan x=\sin x$$

Andrew Bassila

Numerade Educator

Problem 13

Verify the equation is an identity using multiplication and fundamental identities.
$$\sec ^{2} x \cot ^{2} x=\csc ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 14

Verify the equation is an identity using multiplication and fundamental identities.
$$\csc ^{2} x \tan ^{2} x=\sec ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 15

Verify the equation is an identity using multiplication and fundamental identities.
$$\cos x(\sec x-\cos x)=\sin ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 16

Verify the equation is an identity using multiplication and fundamental identities.
$$\tan x(\cot x+\tan x)=\sec ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 17

Verify the equation is an identity using multiplication and fundamental identities.
$$\sin x(\csc x-\sin x)=\cos ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 18

Verify the equation is an identity using multiplication and fundamental identities.
$$\cot x(\tan x+\cot x)=\csc ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 19

Verify the equation is an identity using multiplication and fundamental identities.
$$\tan x(\csc x+\cot x)=\sec x+1$$

Andrew Bassila

Numerade Educator

Problem 20

Verify the equation is an identity using multiplication and fundamental identities.
$$\cot x(\sec x+\tan x)=\csc x+1$$

Andrew Bassila

Numerade Educator

Problem 21

Verify the equation is an identity using factoring and fundamental identities.
$$\tan ^{2} x \csc ^{2} x-\tan ^{2} x=1$$

Andrew Bassila

Numerade Educator

Problem 22

Verify the equation is an identity using factoring and fundamental identities.
$$\sin ^{2} x \cot ^{2} x+\sin ^{2} x=1$$

Andrew Bassila

Numerade Educator

Problem 23

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{\sin x \cos x+\sin x}{\cos x+\cos ^{2} x}=\tan x$$

Andrew Bassila

Numerade Educator

Problem 24

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{\sin x \cos x+\cos x}{\sin x+\sin ^{2} x}=\cot x$$

Andrew Bassila

Numerade Educator

Problem 25

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{1+\sin x}{\cos x+\cos x \sin x}=\sec x$$

Andrew Bassila

Numerade Educator

Problem 26

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{1+\cos x}{\sin x+\cos x \sin x}=\csc x$$

Andrew Bassila

Numerade Educator

Problem 27

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{\sin x \tan x+\sin x}{\tan x+\tan ^{2} x}=\cos x$$

Andrew Bassila

Numerade Educator

Problem 28

Verify the equation is an identity using factoring and fundamental identities.
$$\frac{\cos x \cot x+\cos x}{\cot x+\cot ^{2} x}=\sin x$$

Andrew Bassila

Numerade Educator

Problem 29

Verify the equation is an identity using special products and fundamental identities.
$$\frac{(\sin x+\cos x)^{2}}{\cos x}=\sec x+2 \sin x$$

Andrew Bassila

Numerade Educator

Problem 30

Verify the equation is an identity using special products and fundamental identities.
$$\frac{(1+\tan x)^{2}}{\sec x}=\sec x+2 \sin x$$

Andrew Bassila

Numerade Educator

Problem 31

Verify the equation is an identity using special products and fundamental identities.
$$(1+\sin x)[1+\sin (-x)]=\cos ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 32

Verify the equation is an identity using special products and fundamental identities.
$$(\sec x+1)[\sec (-x)-1]=\tan ^{2} x$$

Andrew Bassila

Numerade Educator

Problem 33

Verify the equation is an identity using special products and fundamental identities.
$$\frac{(\csc x-\cot x)(\csc x+\cot x)}{\tan x}=\cot x$$

Andrew Bassila

Numerade Educator

Problem 34

Verify the equation is an identity using special products and fundamental identities.
$$\frac{(\sec x+\tan x)(\sec x-\tan x)}{\csc x}=\sin x$$

Andrew Bassila

Numerade Educator

Problem 35

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\cos ^{2} x}{\sin x}+\frac{\sin x}{1}=\csc x$$

Andrew Bassila

Numerade Educator

Problem 36

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\sec \alpha}{1}-\frac{\tan ^{2} \alpha}{\sec \alpha}=\cos \alpha$$

Andrew Bassila

Numerade Educator

Problem 37

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\tan x}{\csc x}-\frac{\sin x}{\cos x}=\frac{\sin x-1}{\cot x}$$

Andrew Bassila

Numerade Educator

Problem 38

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\cot x}{\sec x}-\frac{\cos x}{\sin x}=\frac{\cos x-1}{\tan x}$$

Andrew Bassila

Numerade Educator

Problem 39

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\sec x}{\sin x}-\frac{\csc x}{\sec x}=\tan x$$

Andrew Bassila

Numerade Educator

Problem 40

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\csc x}{\cos x}-\frac{\sec x}{\csc x}=\cot x$$

Andrew Bassila

Numerade Educator

Problem 41

Write the given function entirely in terms of the second function indicated.
$$\tan x \text { in terms of } \sin x$$

Andrew Bassila

Numerade Educator

Problem 42

Write the given function entirely in terms of the second function indicated.
$\tan x$ in terms of $\sec x$

Andrew Bassila

Numerade Educator

Problem 43

Write the given function entirely in terms of the second function indicated.
$$\sec x \text { in terms of } \cot x$$

Andrew Bassila

Numerade Educator

Problem 44

Write the given function entirely in terms of the second function indicated.
sec $x$ in terms of $\sin x$

Andrew Bassila

Numerade Educator

Problem 45

Write the given function entirely in terms of the second function indicated.
$$\cot x \text { in terms of } \sin x$$

Andrew Bassila

Numerade Educator

Problem 46

Write the given function entirely in terms of the second function indicated.
$$\cot x \text { in terms of } \csc x$$

Andrew Bassila

Numerade Educator

Problem 47

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\cos \theta=-\frac{20}{29} \text { with } \theta \text { in QII }$$

Andrew Bassila

Numerade Educator

Problem 48

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\sin \theta=\frac{12}{37} \text { with } \theta \text { in QII }$$

Andrew Bassila

Numerade Educator

Problem 49

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\tan \theta=\frac{15}{8} \text { with } \theta \text { in QIII }$$

Andrew Bassila

Numerade Educator

Problem 50

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\sec \theta=\frac{45}{27} \text { with } \theta \text { in QIV }$$

Andrew Bassila

Numerade Educator

Problem 51

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\cot \theta=\frac{x}{5} \text { with } \theta \text { in } \mathrm{QI}$$

Andrew Bassila

Numerade Educator

Problem 52

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\csc \theta=\frac{7}{x} \text { with } \theta \text { in QII }$$

Andrew Bassila

Numerade Educator

Problem 53

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\sin \theta=-\frac{7}{13} \text { with } \theta \text { in QIII }$$

Andrew Bassila

Numerade Educator

Problem 54

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\cos \theta=\frac{23}{25} \text { with } \theta \text { in QIV }$$

Andrew Bassila

Numerade Educator

Problem 55

For the function $f(\theta)$ and the quadrant in which $\theta$ terminates, state the value of the other five trig functions.
$$\sec \theta=-\frac{9}{7} \text { with } \theta \text { in QII }$$

Andrew Bassila

Numerade Educator

Problem 56

Show that the following equations are not identities.
$$\sin \left(\theta+\frac{\pi}{3}\right)=\sin \theta+\sin \left(\frac{\pi}{3}\right)$$

Andrew Bassila

Numerade Educator

Problem 57

Show that the following equations are not identities.
$$\cos \left(\frac{\pi}{4}\right)+\cos \theta=\cos \left(\frac{\pi}{4}+\theta\right)$$

Andrew Bassila

Numerade Educator

Problem 58

Show that the following equations are not identities.
$$\cos (2 \theta)=2 \cos \theta$$

Andrew Bassila

Numerade Educator

Problem 59

Show that the following equations are not identities.
$$\tan (2 \theta)=2 \tan \theta$$

Andrew Bassila

Numerade Educator

Problem 60

Show that the following equations are not identities.
$$\tan \left(\frac{\theta}{4}\right)=\frac{\tan \theta}{\tan 4}$$

Andrew Bassila

Numerade Educator

Problem 61

Show that the following equations are not identities.
$$\cos ^{2} \theta-\sin ^{2} \theta=-1$$

Andrew Bassila

Numerade Educator

Problem 62

Show that the following equations are not identities.
$$\sqrt{\sin ^{2} x-9}=\sin x-3$$

Andrew Bassila

Numerade Educator

Problem 63

The illuminance of a point on a surface by a source of light: $E=\frac{I \cos \theta}{r^{2}}$.
The illuminance $E$ (in lumens/m $^{2}$ ) of a point on a horizontal surface is given by the formula shown, where $I$ is the intensity of the light source in lumens, $r$ is the distance in meters from the light source to the point, and $\theta$ is the complement of the angle $\alpha$ (in degrees) made by the light source and the horizontal surface. Calculate the illuminance if $I=800$ lumens, and the flashlight is held so that the distance $r$ is $2 \mathrm{m}$ while the angle $\alpha$ is $40^{\circ}$.
CAN'T COPY THE FIGURE

Andrew Bassila

Numerade Educator

Problem 64

$$\text { The area of regular polygon: } A=\left(\frac{n x^{2}}{4}\right) \frac{\cos \left(\frac{\pi}{n}\right)}{\sin \left(\frac{\pi}{n}\right)}$$
The area of a regular polygon is given by the formula shown, where $n$ represents the number of sides and $x$ is the length of each side.
a. Rewrite the formula in terms of a single trig function.
b. Verify the formula for a square with sides of $8 \mathrm{m}$.
c. Find the area of a dodecagon ( 12 sides) with 10 -in. sides.

Andrew Bassila

Numerade Educator

Problem 65

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\text { Show that } \cos ^{3} x \text { can be written as } \cos x\left(1-\sin ^{2} x\right)$$.

Andrew Bassila

Numerade Educator

Problem 66

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\text { Show that } \tan ^{3} x \text { can be written as } \tan x\left(\sec ^{2} x-1\right)$$.

Andrew Bassila

Numerade Educator

Problem 67

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
Show that $\tan x+\tan ^{3} x$ can be written as $\tan x\left(\sec ^{2} x\right)$.

Andrew Bassila

Numerade Educator

Problem 68

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\text { Show that } \cot ^{3} x \text { can be written as } \cot x\left(\csc ^{2} x-1\right)$$.

Andrew Bassila

Numerade Educator

Problem 69

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\begin{aligned}&\text { Show } \tan ^{2} x \sec x-4 \tan ^{2} x \text { can be factored into }\\&(\sec x-4)(\sec x-1)(\sec x+1)\end{aligned}$$.

Andrew Bassila

Numerade Educator

Problem 70

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\begin{aligned}&\text { Show } 2 \sin ^{2} x \cos x-\sqrt{3} \sin ^{2} x \text { can be factored }\\&\text { into }(1-\cos x)(1+\cos x)(2 \cos x-\sqrt{3})\end{aligned}$$

Andrew Bassila

Numerade Educator

Problem 71

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\begin{aligned}&\text { Show } \cos ^{2} x \sin x-\cos ^{2} x \text { can be factored into }\\&-1(1+\sin x)(1-\sin x)^{2}\end{aligned}$$.

Andrew Bassila

Numerade Educator

Problem 72

Writing a given expression in an alternative form is an idea used at all levels of mathematics. In future classes, it is often helpful to decompose a power into smaller powers (as in writing $A^{3}$ as $A \cdot A^{2}$ ) or to rewrite an expression using known identities so that it can be factored.
$$\begin{aligned}&\text { Show } 2 \cot ^{2} x \csc x+2 \sqrt{2} \cot ^{2} x \text { can be factored }\\&\text { into } 2(\csc x+\sqrt{2})(\csc x-1)(\csc x+1)\end{aligned}$$.

Andrew Bassila

Numerade Educator

Problem 73

The area of a regular polygon that has been circumscribed about a circle of radius $r(\text { see figure })$ is given by the formula $A=n r^{2} \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$,
CAN'T COPY THE GRAPH
where $n$ represents the number of sides.
(a) Rewrite the formula in terms of a single trig function; (b) verify the formula for a square circumscribed about a circle with radius $4 \mathrm{m} ;$ and (c) find the area of a dodecagon ( 12 sides) circumscribed about the same circle.

Andrew Bassila

Numerade Educator

Problem 74

The perimeter of a regular polygon circumscribed about a circle of radius $r$ is given by the formula $P=2 n r \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$.
where $n$ represents the number of sides. (a) Rewrite the formula in terms of a single trig function;
(b) verify the formula for a square circumscribed about a circle with radius $4 \mathrm{m} ;$ and (c) Find the perimeter of a dodecagon (12 sides) circumscribed about the same circle.

Andrew Bassila

Numerade Educator

Problem 75

At their point of intersection, the angle $\theta$ between any two nonparallel lines satisfies the relationship $\left(m_{2}-m_{1}\right) \cos \theta=$ $\sin \theta+m_{1} m_{2} \sin \theta,$ where $m_{1}$ and $m_{2}$ represent the slopes of the two lines. Rewrite the equation in terms of a single trig function.

Andrew Bassila

Numerade Educator

Problem 76

Use the result of Exercise 75 to find the angle between the lines $Y_{1}=\frac{2}{5} x-3$ and $Y_{2}=\frac{7}{3} x+1$.

Andrew Bassila

Numerade Educator

Problem 77

Use the result of Exercise 75 to find the angle between the lines $Y_{1}=3 x-1$ and $Y_{2}=-2 x+7$.

Andrew Bassila

Numerade Educator

Problem 78

The word tangent literally means "to touch," which in mathematics we take to mean touches in only and exactly one point. In the figure, the circle has a radius of 1 and the vertical line is "tangent" to the circle at the $x$ -axis. The figure can be used to verify the Pythagorean identity for sine and cosine, as well as the ratio identity for tangent. Discuss/Explain how.
CAN'T COPY THE GRAPH

Andrew Bassila

Numerade Educator

Problem 79

Use factoring and fundamental identities to help find the $x$ -intercepts of $f$ in $[0,2 \pi)$.
$$f(\theta)=-2 \sin ^{4} \theta+\sqrt{3} \sin ^{3} \theta+2 \sin ^{2} \theta-\sqrt{3} \sin \theta$$

Andrew Bassila

Numerade Educator

Problem 80

Solve for $x:$
$2351=\frac{2500}{1+e^{-1.015 x}}$

Andrew Bassila

Numerade Educator

Problem 81

Standing 265 ft from the base of the Strastosphere Tower in Las Vegas, Nevada, the angle of elevation to the top of the tower is about $77^{\circ} .$ Approximate the height of the tower to the nearest foot.

Andrew Bassila

Numerade Educator

Problem 82

Use the rational zeroes theorem and other "tools" to find all zeroes of the function $f(x)=2 x^{4}+9 x^{3}-4 x^{2}-36 x-16$.

Andrew Bassila

Numerade Educator

Problem 83

Use a reference rectangle and the rule of fourths to sketch the graph of $y=2 \sin (2 t)$ for $t$ in $[0,2 \pi)$.

Andrew Bassila

Numerade Educator