Unformatted text preview: OER Math 1060 – Trigonometry
Spring 2017 Salt Lake Community College
University of Utah Acknowledgements
Content is based on College Trigonometry, 3rd Corrected Edition by Carl Stitz and
Jeffrey Zeager, to whom we are grateful beyond words for their dedication in
creating and sharing their OER textbook.
This textbook has been developed by Ruth Trygstad, Salt Lake Community
College (SLCC), with contributions from Shawna Haider (SLCC), Spencer
Bartholomew (SLCC) and Maggie Cummings (University of Utah). Many
additional faculty and staff from both Salt Lake Community College and the
University of Utah have helped with this pilot edition. The project has been
sponsored and supported by SLCC, the SLCC Math Department and the University
of Utah Math Department. Special thanks goes to Jason Pickavance, Suzanne
Mozdy and Peter Trapa for their encouragement.
Additional content, including Exercises, has been borrowed from OpenStax
College, Algebra and Trigonometry, openstaxcollege.org/textbooks/collegealgebra-and-trigonometry. Table of Contents
CHAPTER 1 ANGLES AND THEIR MEASURE
1.1 Degree Measure of Angles
1.2 Radian Measure of Angles
1.3 Applications of Radian Measure
CHAPTER 2 THE TRIGONOMETRIC FUNCTIONS
2.1 Right Triangle Trigonometry
2.2 Determining Cosine and Sine Values from the Unit Circle
2.3 The Six Circular Functions
2.4 Verifying Trigonometric Identities
2.5 Beyond the Unit Circle
CHAPTER 3 GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
3.1 Graphs of the Cosine and Sine Functions
3.2 Properties of the Graphs of Sinusoids
3.3 Graphs of the Tangent and Cotangent Functions
3.4 Graphs of the Secant and Cosecant Functions
CHAPTER 4 TRIGONOMETRIC IDENTITIES AND FORMULAS
4.1 The Even/Odd Identities
4.2 The Sum and Difference Identities
4.3 Double Angle Identities
4.4 Power Reduction and Half Angle Formulas 1
191 Table of Contents
4.5 Product to Sum and Sum to Product Formulas
4.6 Using Sum Identities in Determining Sinusoidal Formulas
CHAPTER 5 THE INVERSE TRIGONOMETRIC FUNCTIONS
5.1 Properties of the Inverse Cosine and Sine Functions
5.2 Properties of the Inverse Tangent and Cotangent Functions
5.3 Properties of the Inverse Secant and Cosecant Functions
5.4 Calculators and the Inverse Circular Functions
CHAPTER 6 TRIGONOMETRIC EQUATIONS
6.1 Solving Equations Using the Inverse Trigonometric Functions
6.2 Solving Equations Involving a Single Trigonometric Function
6.3 Solving Equations of Multiple Trigonometric Functions/Arguments
6.3 Exercises 196
271 CHAPTER 7 BEYOND RIGHT TRIANGLES 273 7.1 Solving Triangles with the Law of Sines 274 7.1 Exercises
7.2 Applications of the Law of Sines
7.3 The Law of Cosines
CHAPTER 8 POLAR COORDINATES AND APPLICATIONS
8.1 Polar Coordinates 287
318 8.1 Exercises 334 8.2 Polar Equations 336 8.2 Exercises 342 8.3 Graphing Polar Equations
8.3 Exercises 343
360 Table of Contents
8.4 Polar Representations for Complex Numbers
8.5 Complex Products, Powers, Quotients and Roots
CHAPTER 9 VECTORS AND PARAMETRIC EQUATIONS
9.1 Vector Properties and Operations
9.2 The Unit Vector and Vector Applications 362
410 9.2 Exercises 419 9.3 The Dot Product 422 9.3 Exercises 437 9.4 Sketching Curves Described by Parametric Equations
9.5 Finding Parametric Descriptions for Oriented Curves
9.5 Exercises 440
462 Table of Contents 1 CHAPTER 1
ANGLES AND THEIR MEASURE Chapter Outline
1.1 Degree Measure of Angles
1.2 Radian Measure of Angles
1.3 Applications of Radian Measure
In this chapter we introduce angles in preparation for their critical role in the study of trigonometry.
Beginning with the degree measure of angles in Section 1.1, we move on to radian measure in Section
1.2. The inclusion of conversion between degrees and radians leads to Section 1.3, where radian measure
proves useful in solving applications of circular arc length and area, as well as linear and angular speed.
Throughout this chapter, emphasis remains on angle measure and graphing. In preparation for Chapter 2,
angles are graphed in standard position and coterminal angles are identified. The applications at the end
of Chapter 1 will be the first of many real-world uses to occur throughout our study of trigonometry. 2 Angles and Their Measure 1.1 Degree Measure of Angles
In this section you will: Convert between the Degree-Minute-Second system and decimal degrees. Determine supplementary and complementary angle measures. Graph angles in standard position. Determine coterminal angle measures.
To get started, we recall some basic definitions from geometry. A ray, often described as a half-line, is a
subset of a line that contains a point P along with all points lying to one side of P. The point P from
which the ray originates is called the initial point of the ray, as pictured below. P When two rays share a common initial point they form an angle and the common initial point is called the
vertex of the angle. Following are two examples of angles, the first with vertex R and the second with
vertex S. S
R The two figures below also depict angles. In the first case, the two rays are directly opposite each other
forming what is known as a straight angle. In the second, the rays are identical so the angle is
indistinguishable from the ray itself.
P 1.1 Degree Measure of Angles 3 The measure of an angle is a number which indicates the amount of rotation that separates the rays of the
angle. There is one immediate problem with this, as the following pictures indicate. Which amount of rotation are we attempting to quantify? What we have just discovered is that we have at
least two angles described by this diagram.1 Clearly these two angles have different measures because
one appears to represent a larger rotation than the other, so we must label them differently. In this book,
we use lower case Greek letters such as α (alpha), β (beta), γ (gamma) and θ (theta) to label angles. So,
for instance, we have Degree Measure
One commonly used system to measure angles is degree measure. Quantities measured in degrees are
denoted by a small circle displayed as a superscript. One complete revolution is 360°, and parts of a
revolution are measured proportionately.2 Thus half of a revolution (a straight angle) measures 1
360 180 ; a quarter of a revolution measures 360 90 , and so on. 2
2 The phrase ‘at least’ will be justified in short order.
The choice of 360 is most often attributed to the Babylonians. 4 Angles and Their Measure One revolution ↔ 360° 180° 90° Note that in the previous figure we have used the small square to denote a right angle, as is commonplace
in geometry. If an angle measures strictly between 0° and 90° it is called an acute angle and if it measure
strictly between 90° and 180° it is called an obtuse angle.
We can determine the measure of any angle as long as we know the proportion it represents of an entire
revolution.3 For instance, the measure of an angle which represents a rotation of 2
of a revolution would
3 measure 2
360 240 . The measure of an angle which constitutes only
of a revolution would 3
12 measure 1
360 30 . An angle which indicates no rotation at all is measured as 0°.
12 240° 3 This is how a protractor is graded. 30° 0° 1.1 Degree Measure of Angles 5 Using our definition of degree measure, 1° represents the measure of an angle which constitutes 1
360 revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with
measure smaller than 1°.
There are two ways to subdivide degrees.
1. The first, and most familiar, is decimal degrees. For example, an angle with a measure of 30.5°
would represent a rotation halfway between 30° and 31° or, equivalently 30.5 61 of a full
360 720 rotation.
2. The second way to divide degrees is the Degree-Minute-Second (DMS) system, used in surveying,
global positioning and other applications requiring measurements of longitude and latitude. In the
DMS system, one degree is divided equally into sixty minutes, and in turn each minute is divided
equally into sixty seconds. In symbols, we write 1 60' and 1' 60" , from which it follows that 1 3600" .
EXAMPLE 1.1.1. Convert 42.125° to the DMS system.
SOLUTION. To convert 42.125° to the DMS system, we first note that 42.125 42 0.125 .
Converting the partial amount of degrees to minutes, we find 60' 0.125 7.5'
1 7' 0.5'.
Next, converting the partial amount of minutes to seconds gives 60" 0.5' 30". 1' The result is 42.125 42 0.125 42 7.5' 42 7' 0.5' 42 7' 30" 42 7'30". 6 Angles and Their Measure EXAMPLE 1.1.2. Convert 117°15'45" to decimal degrees.
SOLUTION. To convert 117°15'45" to decimal degrees, we first compute 1 1
15' 60' 4 0.25
and 1 1
45" 3600" 80 0.0125 .
Then we find 117 15'45" 117 15' 45" 117 0.25 0.0125 117.2625 Supplementary and Complementary Angles
Recall that two acute angles are called complementary angles if their measures add to 90°. Two angles,
either a pair of right angles or one acute angle and one obtuse angle, are called supplementary angles if
their measures add to 180°. In the diagrams below, the angles α and β are supplementary angles while the
pair γ and θ are complementary angles. In practice, the distinction between the angle itself and its measure is blurred so that the statement ‘α is an
angle measuring 42°’ is often abbreviated as ‘ 42 ’.
EXAMPLE 1.1.3. Let 111.371 and 37 28'17" .
1. Sketch α and β. 1.1 Degree Measure of Angles 7 2. Find a supplementary angle for α.
3. Find a complementary angle for β.
1. To sketch 111.371 , we first note that 90 180 . If we divide this range in half, we
observe that 90 135 . After one more division, we get 90 112.5 . This gives us a
fairly good estimate for α, as shown in the figure. For 37 28'17" , converting to decimal degrees results in approximately 37.47°. We find 0 90 . After dividing this range in half, we get 0 45 , followed by
22.5 45 , and lastly 33.75 45 . 2. To find a supplementary angle for 111.371 , we seek an angle θ so that 180 . We get 180 180 111.371 68.629 .
3. To find a complementary angle for 37 28'17" , we seek an angle γ so that 90 . We
get 90 90 37 28'17" . While we could reach for the calculator to obtain an
approximate answer, we choose instead to do a bit of sexagesimal4 arithmetic. We first rewrite 4 This is a base-60 system. 8 Angles and Their Measure 90 90 0'0" 89 60'0" 89 59'60".
In essence, we are ‘borrowing’ 1°=60' from the degree place, and then borrowing 1'=60" from the
minutes place.5 This yields 90 37 28'17" 89 59'60" 37 28'17" 52 31'43". Oriented Angles
Up to this point, we have discussed only angles which measure between 0° and 360°, inclusive.
Ultimately, we want to extend their applicability to other real-world phenomena. A first step in this
direction is to introduce the concept of an oriented angle. As its name suggests, in an oriented angle, the
direction of the rotation is important. We imagine the angle being swept out starting from an initial side
and ending at a terminal side, as shown below. When the rotation is counter-clockwise from initial side
to terminal side, we say that the angle is positive; when the rotation is clockwise, we say the angle is
rm i de
ide Initial Side Initial Side A positive angle, 60° A negative angle, –60° We also extend our allowable rotations to include angles which encompass more than one revolution. For
example, to sketch an angle with measure 450° we start with an initial side, rotate counter-clockwise one
complete revolution (to take care of the first 360°), then continue with an additional 90° counterclockwise rotation, as seen below. 5 This is the exact kind of borrowing often taught in elementary school when trying to find 300–125. The difference
is that a base ten system is used to find 300–125; here, it is base sixty. 1.1 Degree Measure of Angles 9 450° Standard Position
To further connect angles with the algebra which has come before, we shall often overlay an angle
diagram on the coordinate plane. An angle is said to be in standard position if its vertex is the origin
and its initial side coincides with the positive x-axis. Angles in standard position are classified according
to where their terminal side lies. For instance, an angle in standard position whose terminal side lies in
Quadrant I is called a Quadrant I angle. If the terminal side of an angle lies on one of the coordinate axes,
it is called a quadrantal angle. Two angles in standard position are called coterminal if they share the
same terminal side.6
In the following figure, 120 and 240 are two coterminal Quadrant II angles drawn in
standard position. Note that 360 , or equivalently 360 . We leave it as an exercise for
the reader to verify that coterminal angles always differ by a multiple of 360°.7 More precisely, if α and β
are coterminal angles, then 360 k where k is an integer.8
y 120 x 240 6 Note that by being in standard position they automatically share the same initial side which is the positive x-axis.
It is worth noting that all of the pathologies of Analytic Trigonometry result from this innocuous fact.
Recall that this means k = 0, ±1, ±2, ···.
7 10 Angles and Their Measure EXAMPLE 1.1.4. Graph each of the (oriented) angles below in standard position and classify them
according to where their terminal side lies. Find three coterminal angles, at least one of which is positive
and one of which is negative.
1. 60 2. 225 3. 540 4. 750 SOLUTION.
1. To graph 60 , we draw an angle with its initial side on the
positive x-axis and rotate counter-clockwise y 60
1 of a
360 6 60 revolution. We see that α is a Quadrant I angle. x To find angles which are coterminal, we look for angles θ of the
form 360 k for some integer k. When k 1 , we get 60 360 420 . Substituting k 1 gives 60 360 300 . If we let k 2 , we get 60 720 780 . 2. Since 225 is negative, we start at the positive x-axis and
rotate clockwise y 225 5 of a revolution. We see that β is a
x Quadrant II angle.
To find coterminal angles, we proceed as before and compute 225 225 360 k for integer values of k. Letting k 1 ,
k 1 and k 2 , we find 135°, –585° and 495° are all coterminal with –225°.
3. Since 540 is positive, we rotate counter-clockwise from the y 540 positive x-axis. One full revolution accounts for 360°, with 180°,
or half of a revolution, remaining. Since the terminal side of γ lies
on the negative x-axis, γ is a quadrantal angle.
All angles coterminal with γ are of the form 540 360 k ,
where k is an integer. Working through the arithmetic, we find
three such angles: 900°, 180° and –180°. x 1.1 Degree Measure of Angles
4. The Greek letter φ is pronounced ‘fee’ or ‘fie’9 and since 11
y 750 is negative, we begin our rotation clockwise from the
positive x-axis. Two full rotations account for 720°, with just 30°
of a revolution to go.
12 x 750 We find that φ is a Quadrant IV angle. To find coterminal angles,
we compute 750 360 k for a few integers k and obtain –390°, –30° and 330°. Note that since there are infinitely many integers, any given angle has infinitely many coterminal angles,
and the reader is encouraged to plot the few sets of coterminal angles found in Example 1.1.4 to see this. The symbol φ represents the small Greek letter phi. We will occasionally use the symbol ϕ to represent the
uppercase Greek letter phi.
9 12 Angles and Their Measure 1.1 Exercises
In Exercises 1 – 4, convert the angles into the DMS system. Round each of your answers to the nearest
1. 63.75 2. 200.325 3. 317.06 4. 179.999 In Exercises 5 – 8, convert the angles into decimal degrees. Round each of your answers to three decimal
5. 125 50' 6. 32 10'12" 7. 502 35' 8. 237 58'43" In Exercises 9 – 12, sketch the angles using the technique presented in Example 1.1.3.
9. 100.491 10. 39.273 11. 172 5'3" 12. 82 9'27" In Exercises 13 – 16, find a supplementary angle for each given angle.
13. 100.491 14. 39.273 15. 172 5'3" 16. 82 9'27" In Exercises 17 – 20, find a complementary angle for each given angle.
17. 10.491° 18. 39.273 19. 27 5'3" 20. 82 9'27" In Exercises 21 – 28, graph the oriented angle in standard position. Classify each angle according to
where its terminal side lies and then give two coterminal angles, one of which is positive and the other
21. 330 22. 135 23. 120 24. 405 25. 270 26. 300 27. 150 28. 135 1.2 Radian Measure of Angles 13 1.2 Radian Measure of Angles
In this section you will: Graph angles in standard position. Determine coterminal angle measures. Convert between degree and radian measures.
The number π is a mathematical constant. That is, it doesn’t matter which circle is selected, the ratio of
its circumference to its diameter will have the same value as any other circle.
The real number π is defined to be the ratio of a circle’s circumference to its diameter. In symbols,
given a circle of circumference C and diameter d, C
d Since the diameter of a circle is twice its radius, we can quickly rearrange the equation formula more useful for our purposes, namely 2 C
to get a
. This tells us that for any circle, the ratio of its
r circumference to its radius is also always constant and that the constant is 2π. Radian Measure
Suppose now we take a portion of the circle, so instead of comparing the entire circumference C to the
radius r, we compare some arc measuring s units in length to the radius r, as depicted below. Let θ be the
central angle subtended by this arc; that is, an angle whose vertex is the center of the circle and whose
determining rays pass through the endpoints of the arc. Using proportionality arguments, it stands to
reason that the ratio ratio, s
should also be a constant among all circles with the same central angle θ. This
, defines the radian measure of an angle.
r 14 Angles and Their Measure s
r r The radian measure of an angle is s
r To get a better feel for radian measure, we note that An angle with radian measure 1 means the corresponding arc length s equals the radius r of the circle.
Hence sr. When the radian measure is 2, we have When the radian measure is 3, s 2r . s 3r , and so forth. Thus, the radian measure of an angle θ tells us how many radius lengths we need to sweep out along the
circle to subtend the angle θ.
r r r r r r
r α has radian measure 1 r β has radian measure 4 Since one revolution sweeps out the entire 2πr, one revolution has radian measure 2 r 2 . From this
r we can find the radian measure of other central angles using proportions, just like we did with degrees.
For instance, half of a revolution has radian measure measure 1 2 , and so forth.
2 1 2 ; a quarter revolution has radian
2 1.2 Radian Measure of Angles 15 Note that, by definition, the radian measure of an angle is a length divided by another length so that these
measurements are actually dimensionless and are considered pure numbers. For this reason, we do not
use any symbols to denote radian measure, but we use the word radians to denote these dimensionless
units as needed. For instance, we say one revolution measures 2π radians; half of a revolution measures π
radians, and so forth. As with degree measure, the distinction between the angle itself and its measure is
often blurred in practice, so when we write 2 , we mean ‘θ is an angle which measures radians’.10
2 We extend radian measure to oriented angles, just ...
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