Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

Chapter 6 Summary and Review

Study Guide

KEY TERMS AND CONCEPTS EXAMPLES

SECTION 6.1: TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES

Trigonometric Function Values of an Acute Angle θ $θ$

Let $θ$ be an acute angle of a right triangle. The six trigonometric functions of $θ$ are as follows:

$\begin{matrix} sin θ = \frac{opp}{hyp}, & csc θ = \frac{hyp}{opp}, \\ cos θ = \frac{adj}{hyp}, & sec θ = \frac{hyp}{adj}, \\ tan θ = \frac{opp}{adj}, & cot θ = \frac{adj}{opp} . \end{matrix}$

If $cos α = \frac{3}{8}$ and $α$ is an acute angle, find the other five trigonometric function values of $α$ .

We find the missing length using the Pythagorean equation: $a^{2} + b^{2} = c^{2}$ .

$\begin{matrix} \begin{matrix} a^{2} + 3^{2} & = & 8^{2} \\ a^{2} & = & 64 - 9 \\ a & = & \sqrt{55} \end{matrix} \\ sin α = \frac{\sqrt{55}}{8}, & csc α = \frac{8}{\sqrt{55}}, or \frac{8 \sqrt{55}}{55}, \\ cos α = \frac{3}{8}, & sec α = \frac{8}{3}, \\ tan α = \frac{\sqrt{55}}{3}, & cot α = \frac{3}{\sqrt{55}}, or \frac{3 \sqrt{55}}{55} & cot α = \frac{3}{\sqrt{55}}, or \frac{3 \sqrt{55}}{55} \end{matrix}$

Reciprocal Functions

$\begin{matrix} csc θ = \frac{1}{sin θ}, sec θ = \frac{1}{cos θ}, \\ cot θ = \frac{1}{tan θ} \end{matrix}$

Given that $sin β = \frac{5}{13}$ , $cos β = \frac{12}{13}$ , and $tan β = \frac{5}{12}$ , find $csc β$ , $sec β$ , and $cot β$ .

$csc β = \frac{13}{5}, sec β = \frac{13}{12}, cot β = \frac{12}{5}$

Function Values of Special Angles

We often use the function values of $30 °$ , $45 °$ , and $60 °$ . Either the triangles below or the values themselves should be memorized.

	$30 °$	$45 °$	$60 °$
sin	$1 / 2$	$\sqrt{2} / 2$	$\sqrt{3} / 2$
cos	$\sqrt{3} / 2$	$\sqrt{2} / 2$	$1 / 2$
tan	$\sqrt{3} / 3$	1	$\sqrt{3}$

Most calculators can convert $D ° M' S ″$ notation to decimal degree notation and vice versa. Procedures among calculators vary. We also can convert without using a calculator.

Find the exact function value.

$\begin{matrix} \csc 45^{°} & = & \frac{2}{\sqrt{2}}, or \sqrt{2}, & \tan 60^{°} & = & \sqrt{3}, \\ \sin 30^{°} & = & \frac{1}{2}, & \cot 45^{°} & = & 1, \\ \cos 60^{°} & = & \frac{1}{2}, & \sec 30^{°} & = & \frac{2}{\sqrt{3}}, or \frac{2 \sqrt{3}}{3}, \\ \cot 30^{°} & = & \frac{3}{\sqrt{3}}, or \sqrt{3}, & \sin 60^{°} & = & \frac{\sqrt{3}}{2}, \\ \tan 45^{°} & = & 1, & \sec 45^{°} & = & \frac{2}{\sqrt{2}}, or \frac{\sqrt{2}}{2}, \\ \csc 60^{°} & = & \frac{2}{\sqrt{3}}, or \frac{3 \sqrt{3}}{3}, & \cos 45^{°} & = & \frac{\sqrt{2}}{2} \end{matrix}$

Convert $17 ° 42' 35 ″$ to decimal degree notation, rounding the answer to the nearest hundredth of a degree.

$\begin{matrix} 17 ° 42' 3 5^{″} & = & 17 ° + 42' + \frac{35'}{60} \approx 17 ° + 42.5833' \\ \approx & 17 ° + \frac{42.5833 °}{60} \approx 17.71 ° \end{matrix}$

Convert $23.12 °$ to $D ° M' S ″$ notation.

$\begin{matrix} 23.12 ° & = & 23 ° + 0.12 \times 1 ° = 23 ° + 0.12 \times 60' \\ = & 23 ° + 7.2' = 23 ° + 7' + 0.2 \times 1' \\ = & 23 ° + 7' + 0.2 \times 6 0^{″} = 23 ° + 7' + 1 2^{″} \\ = & 23 ° 7' 1 2^{″} \end{matrix}$

Cofunction Identities

$\begin{matrix} sin θ = cos (90 ° - θ), \\ cos θ = sin (90 ° - θ), \\ tan θ = cot (90 ° - θ), \\ cot θ = tan (90 ° - θ), \\ sec θ = csc (90 ° - θ), \\ csc θ = sec (90 ° - θ) \end{matrix}$

Given that $sin 47 ° \approx 0.7314$ , $cos 47 ° \approx 0.6820$ , and $tan 47 ° \approx 1.0724$ , find the six trigonometric function values for $43 °$ .

First, we find $csc 47 °$ , $sec 47 °$ , and $cot 47 °$ :

$\begin{matrix} csc 47 ° = \frac{1}{sin 47 °} \approx 1.3672, \\ sec 47 ° = \frac{1}{cos 47 °} \approx 1.4663, \\ cot 47 ° = \frac{1}{tan 47 °} \approx 0.9325. \end{matrix}$

We know that $43 ° = 90 ° - 47 °$ , so we have

$\begin{matrix} sin 43 ° = cos 47 ° \approx 0.6820, \\ cos 43 ° = sin 47 ° \approx 0.7314, \\ tan 43 ° = cot 47 ° \approx 0.9325, \\ csc 43 ° = sec 47 ° \approx 1.4663, \\ sec 43 ° = csc 47 ° \approx 1.3672, \\ cot 43 ° = tan 47 ° \approx 1.0724. \end{matrix}$

SECTION 6.2: APPLICATIONS OF RIGHT TRIANGLES

Solving a Triangle

To solve a triangle means to find the lengths of all sides and the measures of all angles.

Solve this right triangle.

$\begin{array}{l} A & = & ?, & a & = & ?, \\ B & = & 27.3 °, & b & = & 11.6, \\ C & = & 90 °, & c & = & ? \end{array}$

First, we find $A$ : $A = 90 ° - 27.3 ° = 62.7 ° .$

Then we use the tangent and the cosine functions to find $a$ and $c$ :

$\begin{matrix} \begin{matrix} tan 62.7 ° & = & \frac{a}{11.6} \\ 11.6 tan 62.7 ° & = & a \\ 22.5 & \approx & a, \end{matrix} & \begin{matrix} cos 62.7 ° & = & \frac{11.6}{c} \\ c & = & \frac{11.6}{cos 62.7 °} \\ c & \approx & 25.3. \end{matrix} \end{matrix}$

SECTION 6.3: TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

Coterminal Angles

If two or more angles have the same terminal side, the angles are said to be coterminal.

To find angles coterminal with a given angle, we add or subtract multiples of $360 °$ .

Find two positive angles and two negative angles that are coterminal with $123 °$ .

$\begin{matrix} 123 ° & + & 360 ° = 483 °, \\ 123 ° & + & 3 (360 °) = 1203 °, \\ 123 ° & - & 360 ° = - 237 °, \\ 123 ° & - & 2 (360 °) = - 597 ° \end{matrix}$

The angles $483 °$ , $1203 °$ , $- 237 °$ , and $- 597 °$ are coterminal with $123 °$ .

Complementary and Supplementary Angles

Two acute angles are complementary if their sum is $90 °$ .

Two positive angles are supplementary if their sum is $180 °$ .

Find the complement and the supplement of $83.5 °$ .

$\begin{matrix} 90 ° - 83.5 ° & = & 6.5 °, \\ 180 ° - 83.5 ° & = & 96.5 ° \end{matrix}$

The complement of $83.5 °$ is $6.5 °$ , and the supplement of $83.5 °$ is $96.5 °$ .

Trigonometric Functions of Any Angle $θ$

If $P (x, y)$ is any point on the terminal side of any angle $θ$ in standard position, and $r$ is the distance from the origin to $P (x, y)$ , where $r = \sqrt{x^{2} + y^{2}}$ , then

$\begin{array}{l} sin θ & = & \frac{y}{r}, & csc θ & = & \frac{r}{y}, \\ cos θ & = & \frac{x}{r}, & sec θ & = & \frac{r}{x}, \\ tan θ & = & \frac{y}{x}, & cot θ & = & \frac{x}{y} . \end{array}$

The trigonometric function values of $θ$ depend only on the angle, not on the choice of the point on the terminal side that is used to compute them.

Signs of Function Values

The signs of the function values depend only on the coordinates of the point $P$ on the terminal side of an angle.

Find the six trigonometric function values for the angle shown.

We first determine $r$ :

$r = \sqrt{x^{2} + y^{2}} = \sqrt{3^{2} + {(- 4)}^{2}} = \sqrt{25} = 5 .$

$\begin{array}{l} sin θ & = & - \frac{5}{4}, & csc θ & = & - \frac{5}{4}, \\ cos θ & = & \frac{3}{5}, & sec θ & = & \frac{5}{3}, \\ tan θ & = & \frac{4}{3}, & cot θ & = & - \frac{3}{4} . \end{array}$

Given that $cos α = - \frac{1}{5}$ and $α$ is in the third quadrant, find the other function values.

One leg of the reference triangle has length 1, and the length of the hypotenuse is 5. The length of the other leg is $\sqrt{5^{2} - 1^{2}}$ , or $\sqrt{24}$ , or $2 \sqrt{6}$ .

$\begin{array}{l} sin α & = & - \frac{2 \sqrt{6}}{5}, & csc α & = & - \frac{5}{4}, \\ cos α & = & - \frac{1}{5}, & sec α & = & - 5, \\ tan α & = & \frac{4}{3}, or & cot α & = & - \frac{1}{2 \sqrt{6}}, or \frac{\sqrt{16}}{12} \end{array}$

Trigonometric Function Values of Quadrantal Angles

An angle whose terminal side falls on one of the axes is a quadrantal angle.

	$0 °$ $360 °$	$90 °$	$180 °$	$270 °$
sin	0	1	0	$- 1$
cos	1	0	$- 1$	0
tan	0	Not defined	0	Not defined

Find the exact function value.

$\begin{matrix} tan (- 90 °) is not defined, \\ sin 450 ° = 1, \\ csc 270 ° = - 1, \\ cos 720 ° = 1, \\ sec (- 180 °) = - 1, \\ cot (- 360 °) is not defined \end{matrix}$

Reference Angles

The reference angle for an angle is the acute angle formed by the terminal side of the angle and the $x$ -axis.

When the reference angle is $30 °$ , $45 °$ , or $60 °$ , we can mentally determine trigonometric function values.

Find the sine, cosine, and tangent values for $240 °$ .

The reference angle is $240 ° - 180 °$ , or $60 °$ . Recall that $sin 60 ° = \frac{\sqrt{3}}{2}$ , $cos 60 ° = \frac{1}{2}$ , and $tan 60 ° = \sqrt{3}$ .

In the third quadrant, the sine and the cosine functions are negative, and the tangent function is positive. Thus,

$sin 240 ° = - \frac{\sqrt{3}}{2}, cos 240 ° = - \frac{1}{2}, and tan 240 ° = \sqrt{3} .$

Trigonometric Function Values of Any Angle

Using a calculator, we can approximate the trigonometric function values of any angle.

Find each of the following function values using a calculator set in DEGREE mode. Round the values to four decimal places, where appropriate.

$\begin{array}{l} csc 285 ° \approx - 1.0353, & cos 51 ° \approx 0.6293, \\ sin 25 ° 14' 38^{″} \approx 0.4265, & sec (- 45 °) \approx 1.4142, \\ tan (- 1020 °) \approx 1.7321, & sin 810 ° = 1 \end{array}$

Given $cos θ \approx - 0.9724, 180 ° < θ < 270 °$ , find $θ$ .

A calculator shows that the acute angle whose cosine is 0.9724 is approximately $13.5 °$ . We then find angle $θ$ :

$180 ° + 13.5 ° = 193.5 ° .$

Thus, $θ \approx 193.5 °$ .

Aerial Navigation

In aerial navigation, directions are given in degrees clockwise from north. For example, a direction, or bearing, of $195 °$ is shown below.

An airplane flies 320 mi from an airport in a direction of $305 °$ . How far north of the airport is the plane then? How far west?

The distance north of the airport $a$ and the distance west of the airport $b$ are parts of a right triangle. The reference angle is $305 ° - 270 ° = 35 °$ . Thus,

$\begin{matrix} \frac{a}{320} & = & sin 35 ° \\ a & = & 320 sin 35 ° \approx 184; \\ \frac{b}{320} & = & cos 35 ° \\ b & = & 320 cos 35 ° \approx 262. \end{matrix}$

The airplane is about 184 mi north and about 262 mi west of the airport.

SECTION 6.4: RADIANS, ARC LENGTH, AND ANGULAR SPEED

The Unit Circle

A circle centered at the origin with a radius of length 1 is called a unit circle. Its equation is $x^{2} + y^{2} = 1$ .

The circumference of a circle of radius $r$ is $2 π r$ . For a unit circle, where $r = 1$ , the circumference is $2 π$ . If a point starts at $A$ and travels around the circle, it travels a distance of $2 π$ .

Find two real numbers between $- 2 π$ and $2 π$ that determine each of the labeled points.

$\begin{matrix} M : & \frac{5 π}{6}, - \frac{7 π}{6} \\ N : & \frac{3 π}{4}, - \frac{5 π}{4} \\ P : & \frac{π}{2}, - \frac{3 π}{2} \\ Q : & \frac{π}{3}, - \frac{5 π}{3} \\ R : & \frac{11 π}{6}, - \frac{π}{6} \\ S : & \frac{7 π}{4}, - \frac{π}{4} \\ T : & \frac{4 π}{3}, - \frac{2 π}{3} \\ U : & \frac{5 π}{4}, - \frac{3 π}{4} \\ V : & π, - π \end{matrix}$

Radian Measure

Consider the unit circle $(r = 1)$ and arc length 1. If a ray is drawn from the origin through $T$ , an angle of 1 radian is formed. One radian is approximately $57.3 °$ .

A complete counterclockwise revolution is an angle whose measure is $2 π$ radians, or about 6.28 radians. Thus a rotation of $360 °$ (1 revolution) has a measure of $2 π$ radians.

Radian–Degree Equivalents

Converting Between Degree Measure and Radian Measure

To convert from degree measure to radian measure, multiply by $\frac{π radians}{180 °}$ .

To convert from radian measure to degree measure, multiply by $\frac{180 °}{π radians}$ .

If no unit is given for a rotation, the rotation is understood to be in radians.

Convert $150 °$ and $- 63.5 °$ to radian measure. Leave answers in terms of $π$ .

$\begin{matrix} 150 ° = 150 ° \cdot \frac{π radians}{180 °} = \frac{150 °}{180 °} π radians = \frac{5 π}{6}; \\ - 63.5 ° = - 63.5 ° \cdot \frac{π radians}{180 °} = - \frac{63.5 °}{180 °} π radians \approx - 0.35 π \end{matrix}$

Convert $- 328 °$ and $29.2 °$ to radian measure. Round the answers to two decimal places.

$\begin{matrix} - 328 ° = - 328 ° \cdot \frac{π radians}{180 °} = - \frac{328 °}{180 °} π radians \approx - 5.72; \\ 29.2 ° = 29.2 ° \cdot \frac{π radians}{180 °} = \frac{29.2 °}{180 °} π radians \approx 0.51 \end{matrix}$

Convert $- \frac{2 π}{3}$ , $5 π$ , and $- 1.3$ to degree measure. Round the answers to two decimal places.

$\begin{matrix} - \frac{2 π}{3} = - \frac{2 π}{3} \cdot \frac{180 °}{π radians} = - \frac{2}{3} \cdot 180 ° = - 120 °; \\ 5 π = 5 π \cdot \frac{180 °}{π radians} = 5 \cdot 180 ° = 900 °; \\ - 1.3 = - 1.3 \cdot \frac{180 °}{π radians} = \frac{- 1.3 (180 °)}{π} \approx - 74.48 ° \end{matrix}$

Find a positive angle and a negative angle that are coterminal with $\frac{7 π}{4}$ .

$\begin{matrix} \frac{7 π}{4} + 2 π = \frac{7 π}{4} + \frac{8 π}{4} = \frac{15 π}{4}; \\ \frac{7 π}{4} - 3 (2 π) = \frac{7 π}{4} - 6 π = \frac{7 π}{4} - \frac{24 π}{4} = - \frac{17 π}{4} \end{matrix}$

Two angles coterminal with $\frac{7 π}{4}$ are $\frac{15 π}{4}$ and $- \frac{17 π}{4}$ .

Find the complement and the supplement of $\frac{π}{8}$ .

$\begin{matrix} \frac{π}{2} - \frac{π}{8} = \frac{4 π}{8} - \frac{π}{8} = \frac{3 π}{8}; 90 ° = \frac{π}{2} \\ π - \frac{π}{8} = \frac{8 π}{8} - \frac{π}{8} = \frac{7 π}{8} 180 ° = π \end{matrix}$

The complement of $\frac{π}{8}$ is $\frac{3 π}{8}$ , and the supplement of $\frac{π}{8}$ is $\frac{7 π}{8}$ .

Radian Measure

The radian measure $θ$ of a rotation is the ratio of the distance $s$ traveled by a point at a radius $r$ from the center of rotation to the length of the radius r :

$θ = \frac{s}{r} .$

When the formula $θ = s / r$ is used, $θ$ must be in radians and $s$ and $r$ must be expressed in the same unit.

Find the measure of a rotation in radians when a point 6 cm from the center of rotation travels 13 cm.

$θ = \frac{s}{r} = \frac{13 cm}{6 cm} = \frac{13}{6} radians$

Find the length of an arc of a circle of radius 10 yd associated with an angle of $5 π / 4$ radians.

$\begin{matrix} θ = \frac{s}{r}, or s = r θ; \\ s = r θ = 10 yd \cdot \frac{5 π}{4} \approx 39.3 yd \end{matrix}$

Linear Speed and Angular Speed

Linear speed $v$ is the distance $s$ traveled per unit of time $t$ :

$v = \frac{s}{t} .$

Angular speed $ω$ is the amount of rotation $θ$ per unit of time $t$ :

$ω = \frac{θ}{t} .$

Linear Speed in Terms of Angular Speed

The linear speed $v$ of a point a distance $r$ from the center of rotation is given by

$v = r ω,$

where $ω$ is the angular speed, in radians, per unit of time. The unit of distance for $v$ and $r$ must be the same, $ω$ must be in radians per unit of time, and $v$ and $ω$ must be expressed in the same unit of time.

A wheel with a 40-cm radius is rotating at a rate of $2.5 radians/sec$ . What is the linear speed of a point on its rim, in meters per minute?

$\begin{matrix} r = 40 cm \cdot \frac{1 m}{100 cm} = 0.4 m; \\ ω = \frac{2.5 radians}{1 sec} \cdot \frac{60 sec}{1 min} = \frac{150 radians}{1 min}; \\ v = r ω = 0.4 m \cdot \frac{150}{1 min} = \frac{60 m}{min} \end{matrix}$

SECTION 6.5: CIRCULAR FUNCTIONS: GRAPHS AND PROPERTIES

Domains of the Trigonometric Functions

In Sections 6.1 and 6.3, the domains of the trigonometric functions were defined as a set of angles or rotations measured in a real number of degree units. In Section 6.4, the domains were considered to be sets of real numbers, or radians. Radian measure for $θ$ is defined as $θ = s / r$ . When $r = 1, θ = s$ . The arc length $s$ on the unit circle is the same as the radian measure of the angle $θ$ .

Basic Circular Functions

On the unit circle, $s$ can be considered the radian measure of an angle or the measure of an arc length. In either case, it is a real number. Trigonometric functions with domains composed of real numbers are called circular functions.

For a real number $s$ that determines a point $(x, y)$ on the unit circle:

$\begin{matrix} sin s & = & y, \\ cos s & = & x, \\ tan s & = & \frac{y}{x}, x \neq 0, \\ csc s & = & \frac{1}{y}, y \neq 0, \\ sec s & = & \frac{1}{x}, x \neq 0, \\ cot s & = & \frac{x}{y}, y \neq 0. \end{matrix}$

Find each function value using coordinates of a point on the unit circle.

$\begin{matrix} sin (- 5 π) = 0, & csc \frac{π}{3} = \frac{2}{\sqrt{3}}, or \frac{2 \sqrt{3}}{3}, \\ cos (- \frac{3 π}{4}) = - \frac{\sqrt{2}}{2}, & sec \frac{π}{6} = \frac{2}{\sqrt{3}}, or \frac{2 \sqrt{3}}{3}, \\ tan \frac{5 π}{2} is not defined, & cot \frac{23 π}{6} = - \sqrt{3}, \\ cos (- \frac{5 π}{6}) = - \frac{\sqrt{3}}{2}, & tan \frac{7 π}{4} = - 1 \end{matrix}$

Find each function value using a calculator set in RADIAN mode. Round the answers to four decimal places, where appropriate.

$\begin{matrix} cos (- 14.7) \approx - 0.5336, & tan \frac{3 π}{2} is not defined, \\ sin \frac{9 π}{5} \approx - 0.5878, & sec 214 \approx 1.0733 \end{matrix}$

Reflections

Because a unit circle is symmetric with respect to the $x$ -axis, the $y$ -axis, and the origin, the coordinates of one point on the unit circle can be used to find coordinates of its reflections.

The point $(\frac{3}{5}, \frac{4}{5})$ is on the unit circle. Find the coordinates of its reflection across (a) the $x$ -axis, (b) the $y$ -axis, and (c) the origin.

Periodic Function

A function $f$ is said to be periodic if there exists a positive constant $p$ such that

$f (s + p) = f (s)$

for all $s$ in the domain of $f$ . The smallest such positive number $p$ is called the period of the function.

Amplitude

The amplitude of a periodic function is defined as one half of the distance between its maximum and minimum function values. It is always positive.

Sine Function

Continuous
Period: $2 π$
Domain: All real numbers
Range: $[- 1, 1]$
Amplitude: 1
Odd: $sin (- s) = - sin s$

Cosine Function

Continuous
Period: $2 π$
Domain: All real numbers
Range: $[- 1, 1]$
Amplitude: 1
Even: $cos (- s) = cos s$

Tangent Function

Period: $π$
Domain: All real numbers except $(π / 2) + k π$ , where $k$ is an integer
Range: All real numbers

Graph the sine, the cosine, and the tangent functions. For graphs of the cosecant, the secant, and the cotangent functions, see pp. 458–459.

Compare the domains of the sine, the cosine, and the tangent functions.

Function	Domain
sine	All real numbers
cosine	All real numbers
tangent	All real numbers except $π / 2 + k π$ , where $k$ is an integer

Compare the ranges of the sine, the cosine, and the tangent functions.

Function	Range
sine	$[- 1, 1]$
cosine	$[- 1, 1]$
tangent	All real numbers

Compare the periods of the six trigonometric functions.

Function	Period
sine, cosine, cosecant, secant	$2 π$
tangent, cotangent	$π$

SECTION 6.6: GRAPHS OF TRANSFORMED SINE AND COSINE FUNCTIONS

Transformations of the Sine Function and the Cosine Function

To graph $y = A sin (B x - C) + D$ and $y = A cos (B x - C) + D$ :

Stretch or shrink the graph horizontally according to $B$ . Reflect across the $y$ -axis if $B < 0. (Period = | \frac{2 π}{B} |)$
Stretch or shrink the graph vertically according to $A$ . Reflect across the $x$ -axis if $A < 0.$ $(Amplitude = | A |)$
Translate the graph horizontally according to $C / B$ . $(Phase shift = \frac{C}{B})$
Translate the graph vertically according to $D$ .

Determine the amplitude, the period, and the phase shift of

$y = - \frac{1}{2} sin (2 x - \frac{π}{2}) + 1$

and sketch the graph of the function.

$\begin{matrix} y & = & - \frac{1}{2} sin (2 x - \frac{π}{2}) + 1 \\ = & - \frac{1}{2} sin [2 (x - \frac{π}{4})] + 1 \end{matrix}$

Amplitude: $| - \frac{1}{2} | = \frac{1}{2}$
Period: $| \frac{2 π}{2} | = π$
Phase shift: $\frac{π / 2}{2} = \frac{π}{4}$

Review Exercises

Determine whether the statement is true or false.

1. Given that $(- a, b)$ is a point on the unit circle and $θ$ is in the second quadrant, then $cos θ$ is $a$ . [6.4], [6.5]
2. Given that $(- c, - d)$ is a point on the unit circle and $θ$ is in the second quadrant, then $tan θ = - \frac{c}{d}$ . [6.4], [6.5]
3. The measure $300 °$ is greater than the measure 5 radians. [6.4]
4. If $sec θ > 0$ and $cot θ < 0$ , then $θ$ is in the fourth quadrant. [6.3]
5. The amplitude of $y = \frac{1}{2} sin x$ is twice as large as the amplitude of $y = sin \frac{1}{2} x$ . [6.6]
6. The supplement of $\frac{9 π}{13}$ is greater than the complement of $\frac{π}{6}$ . [6.4]
7. Find the six trigonometric function values of $θ$ . [6.1]
8. Given that $β$ is acute and sin $β = \frac{\sqrt{91}}{10}$ , find the other five trigonometric function values. [6.1]

Find the exact function value, if it exists.

9. $cos 45 °$ [6.1]
10. $cot 60 °$ [6.1]
11. $cos 495 °$ [6.3]
12. $sin 150 °$ [6.3]
13. $sec (- 270 °)$ [6.3]
14. $tan (- 600 °)$ [6.3]
15. $csc 60 °$ [6.1]
16. $cot (- 45 °)$ [6.3]
17. Convert $22.27 °$ to degrees, minutes, and seconds. Round the answer to the nearest second. [6.1]
18. Convert $47 ° 33' 2 7^{″}$ to decimal degree notation. Round the answer to two decimal places. [6.1]

Find the function value. Round the answer to four decimal places. [6.3]

19. $tan 2184 °$
20. $sec 27.9 °$
21. $cos 18 ° 13' 4 2^{″}$
22. $sin 245 ° 24'$
23. $cot (- 33.2 °)$
24. $sin 556.13 °$

Find $θ$ in the interval indicated. Round the answer to the nearest tenth of a degree. [6.3]

25. $cos θ = - 0.9041, (180 °, 270 °)$
26. $tan θ = 1.0799, (0 °, 90 °)$

Find the exact acute angle $θ$ in degrees, given the function value. [6.1]

27. $sin θ = \frac{\sqrt{3}}{2}$
28. $tan θ = \sqrt{3}$
29. $cos θ = \frac{\sqrt{2}}{2}$
30. $sec θ = \frac{2 \sqrt{3}}{3}$
31. Given that $sin 59.1 ° \approx 0.8581, cos 59.1 ° \approx 0.5135$ , and $tan 59.1 ° \approx 1.6709$ , find the six function values for $30.9 °$ . [6.1]

Solve each of the following right triangles. Standard lettering has been used. [6.2]

32. $a = 7.3, c = 8.6$
33. $a = 30.5, B = 51.17 °$
34. One leg of a right triangle bears east. The hypotenuse is 734 m long and bears $N 57 ° 23' E$ . Find the perimeter of the triangle.
35. An observer’s eye is 6 ft above the floor. A mural is being viewed. The bottom of the mural is at floor level. The observer looks down $13 °$ to see the bottom and up $17 °$ to see the top. How tall is the mural?

For angles of the following measures, state in which quadrant the terminal side lies. [6.3]

36. $142 ° 11' 5^{″}$
37. $- 635.2 °$
38. $- 392 °$

Find a positive angle and a negative angle that are coterminal with the given angle. Answers may vary.

39. $65 °$ [6.3]
40. $\frac{7 π}{3}$ [6.4]

Find the complement and the supplement.

41. $13.4 °$ [6.3]
42. $\frac{π}{6}$ [6.4]
43. Find the six trigonometric function values for the angle $θ$ shown. [6.3]
44. Given that $tan θ = 2 / \sqrt{5}$ and that the terminal side is in quadrant III, find the other five function values. [6.3]
45. An airplane travels at 530 mph for $3 \frac{1}{2}$ hr in a direction of $160 °$ from Minneapolis, Minnesota. At the end of that time, how far south of Minneapolis is the airplane? [6.3]
46. On a unit circle, mark and label the points determined by $7 π / 6, - 3 π / 4, - π / 3$ , and $9 π / 4$ . [6.4]

For angles of the following measures, convert to radian measure in terms of $π$ and convert to radian measure not in terms of $π$ Round the answer to two decimal places. [6.4]

47. $145.2 °$
48. $- 30 °$

Convert to degree measure. Round the answer to two decimal places where appropriate. [6.4]

49. $\frac{3 π}{2}$
50. 3
51. $- 4.5$
52. $11 π$
53. Find the length of an arc of a circle, given a central angle of $π / 4$ and a radius of 7 cm. [6.4]
54. An arc 18 m long on a circle of radius 8 m subtends an angle of how many radians? how many degrees, to the nearest degree? [6.4]
55. A waterwheel in a watermill has a radius of 7 ft and makes a complete revolution in 70 sec. What is the linear speed, in feet per minute, of a point on the rim? [6.4]
56. An automobile wheel has a diameter of 14 in. If the car travels at a speed of 55 mph, what is the angular velocity, in radians per hour, of a point on the edge of the wheel? [6.4]
57. The point $(\frac{3}{5}, - \frac{4}{5})$ is on a unit circle. Find the coordinates of its reflections across the $x$ -axis, the $y$ -axis, and the origin. [6.5]

Find the exact function value, if it exists. [6.5]

58. $cos π$
59. $tan \frac{5 π}{4}$
60. $sin \frac{5 π}{3}$
61. $sin (- \frac{7 π}{6})$
62. $tan \frac{π}{6}$
63. $cos (- 13 π)$

Find the function value, if it exists. Round the answer to four decimal places. [6.5]

64. $sin 24$
65. $cos (- 75)$
66. $cot 16 π$
67. $tan \frac{3 π}{7}$
68. $sec 14.3$
69. $cos (- \frac{π}{5})$
70. Graph each of the six trigonometric functions from $- 2 π$ to $2 π$ . [6.5]
71. What is the period of each of the six trigonometric functions? [6.5]
72. Complete the following table. [6.5]

Function Domain Range

sine

cosine

tangent
73. Complete the following table with the sign of the specified trigonometric function value in each of the four quadrants. [6.3]

Function I II III IV

sine

cosine

tangent

Determine the amplitude, the period, and the phase shift of the function, and sketch the graph of the function. [6.6]

74. $y = sin (x + \frac{π}{2})$
75. $y = 3 + \frac{1}{2} cos (2 x - \frac{π}{2})$

In Exercises 76–79, match the function with one of the graphs (a)–(d) that follow. [6.6]

76. $y = cos 2 x$
77. $y = \frac{1}{2} sin x + 1$
78. $y = - 2 sin \frac{1}{2} x - 3$
79. $y = - cos (x - \frac{π}{2})$
80. Sketch a graph of $y = 3 cos x + sin x$ for values of $x$ between 0 and $2 π$ . [6.6]
81. Graph: $f (x) = e^{- 0.7 x} cos x$ . [6.6]
82. Which of the following is the reflection of $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$ across the $y$ -axis? [6.5]
1. $(\frac{1}{2}, - \frac{\sqrt{3}}{2})$
2. $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
3. $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
4. $(\frac{\sqrt{3}}{2}, - \frac{1}{2})$
83. Which of the following is the domain of the cosine function? [6.5]
1. $(- 1, 1)$
2. $(- \infty, \infty)$
3. $[0, \infty)$
4. $[- 1, 1]$
84. The graph of $f (x) = - cos (- x)$ is which of the following? [6.6]

Synthesis

85. Graph $y = 3 sin (x / 2)$ , and determine the domain, the range, and the period. [6.6]
86. In the following graph, $y_{1} = sin x$ is shown and $y_{2}$ is shown in red. Express $y_{2}$ as a transformation of the graph of $y_{1}$ . [6.6]
87. Find the domain of $y = log (cos x)$ . [6.6]
88. Given that $sin x = 0.6144$ and that the terminal side is in quadrant II, find the other basic circular function values. [6.3]

Collaborative Discussion and Writing

89. Compare the terms radian and degree. [6.1], [6.4]
90. In circular motion with a fixed angular speed, the length of the radius is directly proportional to the linear speed. Explain why with an example. [6.4]
91. Explain why both the sine function and the cosine function are continuous, but the tangent function, defined as $sine/cosine$ , is not continuous. [6.5]
92. In the transformation steps listed in Section 6.6, why must step (1) precede step (3)? Give an example that illustrates this. [6.6]
93. In the equations $y = A sin (B x - C) + D$ and $y = A cos (B x - C) + D$ , which constants translate the graphs and which constants stretch and shrink the graphs? Describe in your own words the effect of each constant. [6.6]
94. Two new cars are each driven at an average speed of 60 mph for an extended highway test drive of 2000 mi. The diameters of the wheels of the two cars are 15 in. and 16 in., respectively. If the cars use tires of equal durability and profile, differing only by the diameter, which car will probably need new tires first? Explain your answer. [6.4]

6 Chapter Test

1. Find the six trigonometric function values of $θ$ .

Find the exact function value, if it exists.

2. $sin 120 °$
3. $tan (- 45 °)$
4. $cos 3 π$
5. $sec \frac{5 π}{4}$
6. Convert $38 ° 27' 5 6^{″}$ to decimal degree notation. Round the answer to two decimal places.

Find the function values. Round the answers to four decimal places.

7. $tan 526.4 °$
8. $sin (- 12 °)$
9. $sec \frac{5 π}{9}$
10. $cos 76.07$
11. Find the exact acute angle $θ$ , in degrees, for which $sin θ = \frac{1}{2}$ .
12. Given that $sin 28.4 ° \approx 0.4756, cos 28.4 ° \approx 0.8796$ , and $tan 28.4 ° \approx 0.5407$ , find the six trigonometric function values for $61.6 °$ .
13. Solve the right triangle with $b = 45.1$ and $A = 35.9 °$ . Standard lettering has been used.
14. Find a positive angle and a negative angle coterminal with a $112 °$ angle.
15. Find the supplement of $\frac{5 π}{6}$ .
16. Given that $sin θ = - 4 / \sqrt{41}$ and that the terminal side is in quadrant IV, find the other five trigonometric function values.
17. Convert $210 °$ to radian measure in terms of $π$ .
18. Convert $\frac{3 π}{4}$ to degree measure.
19. Find the length of an arc of a circle given a central angle of $π / 3$ and a radius of 16 cm.

Consider the function $y = - sin (x - π / 2) + 1$ for Exercises 20–23.

20. Find the amplitude.
21. Find the period.
22. Find the phase shift.
23. Which of the following is the graph of the function?
1. a)
2. b)
3. c)
4. d)
24. Ski Dubai Resort. Ski Dubai is the first indoor ski resort in the Middle East. Its longest ski run drops 60 ft and has an angle of depression of approximately $8.6 °$ (Source: www.SkiDubai.com). Find the length of the ski run. Round the answer to the nearest foot.
25. Location. A motor home travels at 50 mph for 6 hr in a direction of $115 °$ from Flagstaff, Arizona. At the end of that time, how far east of Flagstaff is the motor home?
26. Linear Speed. A ferris wheel has a radius of 6 m and revolves at 1.5 rpm. What is the linear speed, in meters per minute?
27. Graph: $f (x) = \frac{1}{2} x^{2} sin x$ .
28. The graph of $f (x) = - sin (- x)$ is which of the following?