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APPENDIX A: Trigonometry
Basics
Trigonometry Basics A2
Degree Measure A2
Right-Triangle Trigonometry A4
Unit-Circle Trigonometry A8
Radian Measure A10
Angle Measure Conversions
A14
Sine and Cosine Functions
A15
A.1 Concept Inventory
A19
A.1 Activities
A19
Answers to Appendix A
A25
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APPENDIX A
Trigonometry Basics
This appendix gives supplementary material on degree measure and right-triangle
trigonometry. It can be used in connection with Section 8.1.
Degree Measure
One complete revolution of a circle is divided into 360 equal parts called degrees.
Talking about “small parts of a rotation” is awkward, so we call partial rotations
angles. The measure of an angle is described by the amount of rotation in the turn.
For example, a 90-degree angle is one-fourth of a complete counterclockwise revolution (see Figure A.1). A small circle as a superscript on the angle measure is used as a
symbol to indicate degrees (90°).
90°
1
A 90° angle is 4
of a complete revolution.
FIGURE A.1
Degree Measurement
360 degrees 360° 1 complete revolution
1 degree 1° 1
of a complete revolution
360
The selection of 360 as the number of divisions is rooted in history and may
reflect the fact that the Earth completes one revolution about the Sun in approximately
365 days. Because 365 does not have a large number of divisors (only 5 and 73), the
inventors of the system probably picked 360 with its multitude of divisors (2, 3, 4, 5,
6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180) as being close to
the number of days of the year and easy to use in computing. The fortunate choice of
360° gives us a large number of even-degree angles that are simple fractions of a full
revolution.*
*It is possible to divide one revolution into 400 equal parts. Incidentally, some do. These parts are called
gradians or grads, and this angle measure is available on many calculators. Of course, you might prefer to
be especially patriotic and divide one revolution into 1776 equal parts. In this case, you would be inventing a new angle measure.
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Trigonometry Basics
Measuring Angles in Degrees
1
a. Express 8 of a complete rotation in degrees.
1
b. Express 3 of a complete rotation in degrees.
Solution Keeping in mind that one complete rotation is 360°, we have
1
a. 8 (360°) 45° (See Figure A.2a.)
1
b. 3 (360°) 120° (See Figure A.2b.)
120°
45°
1
8 of a revolution
(a)
(b)
is a 45° angle.
1
3 of a revolution
is a 120° angle.
●
FIGURE A.2
In the study of trigonometry, we consider revolutions around a particular circle.
Specifically, we consider the unit circle—that is, the circle with radius 1, centered at
the origin. We say that an angle is in standard position when the vertex of the angle
is at the origin and one of its sides is drawn along the positive x-axis. The side that is
drawn on the positive x-axis is called the initial side. The other side of the angle is
called the terminal side. Refer to Figure A.3.
Figure A.4 shows a 90° angle, a 160° angle, and a 250° angle.
y
1
y
=
Te
250°
us
in
rm
al
e
sid
θ
di
EXAMPLE 1
Ra
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Initial side
FIGURE A.3
160°
90°
x
x
FIGURE A.4
Because a 90° angle is one-quarter of a rotation (one quarter of the way around the
full circle), a 180° angle would be one-half of a rotation, a 270° angle would be
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APPENDIX A
y
-90°
-200°
x
FIGURE A.5
three-quarters of a rotation, and a 360° angle is one full rotation. Note that in Figure
A.4, the 160° angle is drawn between a quarter rotation and a half rotation, and the
250° angle is drawn between a half rotation and a three-quarter rotation. You should
also note that all of the rotations are drawn in the counterclockwise direction. We
consider counterclockwise rotation to define a positive angle and clockwise rotation
to define a negative angle. For example, Figure A.5 illustrates a 90° angle and a 200°
angle.
When angle measures are larger than 360°, they describe more than one rotation
around the circle. For example, an 810° angle describes two full rotations plus onequarter of a rotation in the counterclockwise direction (2 360 90 810). A
505° angle describes one full rotation plus one-quarter of a rotation plus an extra
55° angle in the clockwise direction (360 90 55 505). These two angles are
shown in Figure A.6.
y
y
x
810°
x
-505°
FIGURE A.6
Before defining trigonometric functions for angles on a unit circle, we take a brief
look at the trigonometric functions defined in terms of a right triangle.
use
ten
o
yp
H
θ
Leg adjacent to θ
FIGURE A.7
Leg opposite θ
Right-Triangle Trigonometry
From the early use of trigonometry (meaning “triangle measurement”) through
present-day applications, right triangles have provided a method for solving problems that involve indirect measurements. A right triangle is a triangle with one 90°
angle. In applications, we generally label one of the remaining angles of the right triangle as angle . (See Figure A.7.)
We define the sine of the angle as the ratio of the length of the leg opposite the
angle to the length of the hypotenuse (the side opposite the right angle). The cosine
of the angle is the ratio of the length of the adjacent leg to the length of the
hypotenuse. The tangent of the angle is the ratio of the length of the leg opposite
the angle to the length of the adjacent leg. We abbreviate cosine by cos , sine by
sin , and tangent by tan .
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Trigonometry Basics
Right-Triangle Trigonometric Definitions
For a right triangle with one of the non-90° angles having measure , the
sine, cosine, and tangent of are defined in terms of the sides of the triangle.
sin length of the leg opposite the angle length of the hypotenuse
cos length of the leg adjacent to the angle length of the hypotenuse
tan length of the leg opposite the angle length of the leg adjacent to the angle These three functions are illustrated in Figure A.8.
use
ten
o
yp
H
θ
use
ten
o
yp
H
θ
θ
Adjacent leg
sin θ =
Opposite leg
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opposite leg
hypotenuse
cos θ =
adjacent leg
hypotenuse
Adjacent leg
tan θ =
opposite leg
adjacent leg
FIGURE A.8
Three other functions that are commonly studied in trigonometry are the secant,
cosecant, and cotangent functions. They are defined as follows:
secant :
sec 1
length of the hypotenuse
cos length of the adjacent leg
cosecant :
csc 1
length of the hypotenuse
sin length of the opposite leg
cotangent :
cot 1
length of the adjacent leg
tan length of the opposite leg
We study primarily sin , cos , and tan because the other three trigonometric
functions are defined in terms of these first three.
If we are given any right triangle for which we know the angle and know the
length of one side of the triangle, we can use the trigonometric functions sin , cos ,
and tan to obtain the lengths of the other two sides of the triangle. For example, if
we have a right triangle with a 25° angle and hypotenuse of length 2 inches, we can
use sin 25° and cos 25° to find the lengths of the two legs. (See Figure A.9.)
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APPENDIX A
s
che
2 in
b
25°
a
FIGURE A.9
To find the length of the leg adjacent to the 25° angle, we use the fact that cos 25° is
the ratio of the length of the adjacent leg to the length of the hypotenuse. We will use
a
a to represent the length of the adjacent leg, so cos 25° 2. Solving for a, we have
a 2 cos 25° 2(0.90631) 1.81262 inches
A.1
That is, the leg adjacent to the 25° angle is approximately 1.8 inches long. Similarly,
the length of side b can be found as
b 2 sin 25° 2(0.42262) 0.84524 inch
Trigonometry is often used in applications where indirect measurement of the
length of one or more legs of a triangle is necessary. Land surveying often relies on
the use of an instrument called a sextant to measure angles and on trigonometry to
determine distances using right triangles.
EXAMPLE 2
Using Trig to Determine Measurement
Castle Measurement A soldier in ancient times making use of triangle ratios is
illustrated in Figure A.10. He is using a sextant and sighting it in line with the top of
the castle wall. The sextant gives an angle of 14.6°. The soldier then counts the bricks
in the wall and estimates the distance d without having to enter the battle zone
between his safe spot and the castle wall.
h
θ = 14.6°
d
FIGURE A.10
a. Estimate the height of the wall if each brick is 11 inches tall.
b. Find the distance to the castle.
c. Find the distance from the soldier to the top of the castle wall.
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Trigonometry Basics
Solution
a. There are 22 layers of bricks in the wall, so the wall is (22 bricks)(11 inches per
brick) 242 inches 20.2 feet tall.
b. Using the definition of the tangent of an angle, we know that
tan 14.6° 20.2 feet
d
so we have
d
20.2 feet
20.2 feet
77.4 feet
tan 14.6°
0.26048
The soldier is approximately 77.4 feet away from the castle.
c. The Pythagorean Theorem yields
h 20.22 77.42 6400.7 80 feet
The soldier is approximately 80 feet away from the top of the castle wall.
●
Two special right triangles that occur in applications using trigonometry are the
30°60° right triangle and the 45°45° right triangle. The values of sine, cosine,
and tangent for one of these triangles are explored in the next example. The other triangle is left for an activity.
EXAMPLE 3
60°
2s
30°
s
Determining Ratios for a Special Triangle
The 30°–60° right triangle is so named because in a right triangle with a 30° angle,
the remaining angle must be 60°. A reflection of the triangle across the leg between
the 30° angle and the right angle forms an equilateral triangle (one whose three sides
are of equal length). The line of reflection is the perpendicular bisector of the base.
(See Figure A.11.)
h
a. Use the Pythagorean Theorem to find the length h in terms of s.
s
60°
FIGURE A.11
b. Use the sides h and s to find the sine, cosine, and tangent for the 30° angle in a
30°–60° right triangle.
c. Use the sides h and s to find the sine, cosine, and tangent for the 60° angle.
Solution
a. The side opposite the 30° angle has length s that is equal to half of the length 2s of
the hypotenuse. Applying the Pythagorean Theorem yields
h 2 s 2 (2s)2
Solving for h, we find the remaining leg length to be
h (2s )2 s 2 4s2 s 2 3s 2 3s
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APPENDIX A
2s
b. Referring to Figure A.12, we find the values of sine, cosine, and tangent for the
30° angle in the right triangle.
60°
s
30°
√3s
FIGURE A.12
sin 30° s
1
2s 2
cos 30° 3s 3
2s
2
tan 30° s
1
3s 3
c. Similarly, the values of sine, cosine, and tangent for the 60° angle follow:
sin 60° 3s 3
2s
2
cos 60° s
1
2s 2
tan 60° 3s
3
s
●
Unit-Circle Trigonometry
y
1
θ
x
y
x
Right triangle
associated with angle FIGURE A.13
Right-triangle trigonometry is useful in many applications. However, as we previously noted, the trigonometric functions are not restricted to use with angles that are
less than 90°.
We can now define the trigonometric functions sine, cosine, and tangent for all
angles by referring to our knowledge of right-triangle trigonometry. When the angle
is between 0° and 90°, we draw a right triangle in the unit circle such that the
hypotenuse of the triangle is along the terminal side of the angle from the origin to
the unit circle, and the other two legs of the right triangle are drawn along the x-axis
and perpendicular to the x-axis, as shown in Figure A.13.
If we let (x, y) represent the point where the terminal edge of angle intersects
the unit circle, then the leg drawn along the x-axis has length x and the leg drawn perpendicular to the x-axis has length y. The hypotenuse has length 1 (the radius of the
unit circle). We define the trigonometric functions sin , cos , and tan as we did
for the right triangle, so that
sin y
y
1
x
x
1
y
tan x
cos We define the trigonometric functions similarly for angles that are not between
0° and 90°. First, draw the angle on the unit circle. Then draw a right triangle such
that the hypotenuse of the triangle is along the terminal side of the angle from the
origin to the unit circle, and the other two legs of the right triangle are drawn along
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the x-axis (possibly in the negative x direction) and perpendicular to the x-axis (possibly in the negative y direction) as shown in Figures A.14a and b.
y
y
(x, y) = (-a, b)
b
1
θ
-a
θ
-c
x
-d
x
1
(x, y) = (-c, -d)
(a)
(b)
FIGURE A.14
Once again, call the point where the terminal edge of angle intersects the unit
circle (x, y), and let a represent the length of the leg drawn along the x-axis and let b
represent the length of the leg drawn perpendicular to the x-axis. The hypotenuse has
length 1 (the radius of the unit circle). However, we must indicate whether the legs of
the triangle are drawn in the negative x and/or the negative y direction. We indicate
this by writing the negative of the length in the appropriate direction. For example,
the triangle in Figure A.14a has one leg along the negative portion of the x-axis. This
leg has length a in the negative x direction, so x a. The other leg is drawn up from
the x-axis; thus it has positive direction and has length y h.
We define the trigonometric functions sin , cos , and tan as we have done prey
y
x
viously, so that sin 1 y, cos 1 x, and tan x . Thus, for the angle
b
b
drawn in Figure A.14a, sin b, cos a, and tan a a . Similarly, for the
d
d
angle drawn in Figure A.14b, sin d, cos c, and tan c c .
In general, for any angle , we define sine, cosine, and tangent as follows:
Unit-Circle Trigonometric Definitions
Let (x, y) be the point on the unit circle where the terminal side of the angle intersects the circle. Then the sine, cosine, and tangent of the angle are
sin y
cos x
tan A.2
y
x
Because the unit circle is given by the equation x 2 y 2 1, it is true that for any
angle , cos2 sin2 1.
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APPENDIX A
EXAMPLE 4
Determining Ratios for a Specific Angle
A 420° angle is shown in Figure A.15.
y
a. Draw the right triangle associated with a 420° angle. Label the angle in the triangle between the terminal side and the x-axis.
b. Find the sine, cosine, and tangent of a 420° angle.
-420°
x
Solution
a.
y
FIGURE A.15
60°
x
FIGURE A.16 Right triangle
associated with the 420° angle
b. This right triangle is one of the special triangles mentioned in Example 3. It is a
3
30°-60° right triangle. We know from Example 3 that sin 60° 2 and
1
1
cos 60° 2. Thus the leg on the x-axis is positive and has length cos 60° 2, so
1
x 2. The other leg is drawn down from the x-axis, so it’s measure is negative and
3
y
1
2
1
FIGURE A.17
3
has length sin 60° 2 , so y 2 . Place these values on Figure A.16. (See Figure
A.17.) Now we can read the trigonometric values from the figure:
x
sin(420°) y 3
2
cos(420°) x 1
2
-√3
2
tan(420°) y
3
x
●
We have now defined trigonometric functions as they apply to right triangles
and, in the much broader sense, as they apply to unit circles. However, our discussion
of unit-circle trigonometry would not be complete if we did not consider another
angle measure called radian measure. In fact, we cannot use the trig functions in calculus without considering radian measure of angles.
Radian Measure
Another unit of angle measurement is called a radian. Picture a circular pizza. We can
describe a slice of pizza in terms of the angle of the wedge formed by the slice. For
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Trigonometry Basics
FIGURE A.18
1
6 of
a pizza is a 60° wedge.
A11
instance, in angular terms we would describe one-eighth of a pizza as a 45° wedge. A
one-sixth slice would be described as a 60° wedge (see Figure A.18).
Another way to think of a slice of pizza is in terms of the amount of crust around
the edge. If there were something special in the crust, like a cheese filling, then we
might want to focus on the length of the crust around the circular edge of the slice. In
doing so, we would be talking about an arc of a circle— that is, the slice’s portion of
the circumference. (A portion of the circle is called an arc, and the complete distance
around the circle is its circumference.) If the radius r of the pizza in Figure A.18 were
1 foot, then the circumference of the pizza would be 2r 2(1 foot) 2 feet 6.3 feet. Basic geometry tells us that the arc length is the same fractional part of the
circumference of the circle as the angle is of one complete rotation. Thus the 45°
45°
wedge would form an arc of 360° (2 feet) 0.8 foot of the special cheese crust. The
60°
s
r
The
radian measure of an
angle is the ratio of
the arc length to the
radius.
FIGURE A.19
s
r
60° wedge would form an arc of 360° (2 feet) 1 foot.
We define the radian measure of an angle to be the ratio of the length of the arc s
cut out of the circumference of a circle by the angle to the radius r of the circle. (See
Figure A.19.) Because the total arc length (that is, the circumference) of a circle is 2
2r
times its radius, a full revolution has radian measure of r 2. (See Figure A.20.)
Note that 2 is a real number approximately equal to 6.283. When we are using a
unit circle, the radius is 1, so the radian measure is simply the corresponding arc
s
s
length: r 1 s.
It is important to recognize that the radian measure of an angle is a real number
with no units attached. In the definition of radian measure, both s and r measure
length and must have the same units, which causes to be a unitless quantity. However, we sometimes report this measure as radians.
Radian Measurement
2 radians 1 complete counterclockwise rotation
1
1
1 radian counterclockwise rotation (a little less than 6 revolution)
2
= 2π radians
= 1 radian
(a)
(b)
FIGURE A.20
Figure A.21 shows some common radian measures on a circle.
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APPENDIX A
3π
4
π
2
π
4
π
0, 2π
5π
4
3π
2
7π
4
FIGURE A.21 The angle corre3
sponds to 8 of a complete rotation,
3
3
or 8(2) 4 units around the
3
circle. Thus its radian measure is 4 .
EXAMPLE 5
Understanding Radian Measure of Angles
-5
a. Draw angles with radian measure 4 and 5 on a unit circle.
b. Estimate the radian measure of the two angles shown in Figures A.22a and b.
(a)
(b)
FIGURE A.22
Solution
=5
1
-5
1
a. An angle of 4 4 2(2 ) 8(2 ) is half of
1
a clockwise rotation plus another 8 of a clockwise rotation
= −5π
4
3
(see Figure A.23a). An angle of 5 is between 4 of a counter(a)
(b)
FIGURE A.23
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3
clockwise rotation 2 4.7 and a complete rotation
(2 6.3) (see Figure A.23b).
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Trigonometry Basics
1
2
b. The angle in Figure A.22a appears to be approximately 12 of a rotation, or 12 0.5
radian. The angle in Figure A.22b is 3: one complete clockwise rotation (2)
plus a half of a clockwise rotation (). ●
EXAMPLE 6
Measuring Angles in Radians
1
a. Express 8 of a complete rotation in radians.
1
b. Express 4 of a complete rotation in radians.
c. Express 4 radians in terms of complete rotations.
Solution Keeping in mind that 1 complete rotation is 2 radians, we have
1
1
a. 8(2) 4 radians
b. 4(2) 2 radians
1
1
c. Because 1 radian 2 of a complete revolution, 4 radians is 4 2 2
complete rotations. ●
The odometer on an automobile converts turns of the drive shaft into distances
so that you can observe the number of miles you have traveled. Although the next
example explores this idea, it is not unique to automobiles. Whenever wheels rotate
to cause movement, the same type of relationship between rotations of the wheels
and distance traveled holds true.
EXAMPLE 7
Using Radius to Determine Arc Length
Rotating Wheels Consider a front-wheel-drive automobile with wheels that are
20 inches in diameter.
a. How far will 1 rotation of the wheels cause the automobile to travel?
b. How many times will the wheels revolve when the automobile travels 1 mile?
Solution
a. As an automobile travels, its drive shaft and front tires turn at the same rate; that
is, 1 revolution in one causes 1 revolution in the other. If the wheels are 20 inches
in diameter (radius 10 inches), then each rotation causes the automobile to
move forward 2r (2)(10 inches) 62.8 inches 5.2 feet. Each rotation of
the wheels causes the automobile to travel about 62.8 inches or 5.2 feet.
12 inches
foot 63,360 inches in a mile. So
inches 1 revolution
revolutions
63,360 mile 210 inches 1008.4 mile
feet
b. There are 5280 mile
The wheels must revolve approximately 1008.4 times in each mile traveled.
●
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APPENDIX A
Angle Measure Conversions
Whenever you have two measurement scales that can be applied to the same object, it
is important to have a conversion technique. In this case, 1 revolution is both 360°
2
and 2 radians. Thus degrees are converted to radians by multiplying by 360 180,
180
and radians are converted to degrees by multiplying by . Our conversion formula is
Angle Conversion Formula
radians 180°
Up to this point, we have used the word radian to specify our choice of angle
measure, not the units of the angle. You may have noted that it is cumbersome to
write the word radians each time angle measure is used. Therefore, we adopt the following convention.
Angle Measure Convention
All angles are understood to be measured in radians unless the degree symbol
is used to specify degree measure of the angle.
That is, an angle with measure 60 is quite different than one with measure 60°. An
angle of 60 radians is more than 9 full revolutions, whereas an angle of 60° is only
one-sixth of a revolution. If you mean an angle of 60 degrees, be certain that you use
the degree symbol with the angle.
EXAMPLE 8
Converting Angles
7
a. Express 6 in degrees.
A.3a, b
b. Express 135° in radians.
Solution
a.
b.
210°
3
135° (135°) 180° 4
7
7
6
6
180°
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Trigonometry Basics
Sine and Cosine Functions
We now define sine and cosine functions of angles in standard position on a unit circle. The box also contains a very important trig identity that follows from the fact
that the equation of the unit circle at the origin of the x- and y-axes is x 2 y 2 1.
Trigonometric Values for a General Angle
Consider an angle in standard
position. The terminal side of the
angle intercepts the unit circle at
a point (x, y). The sine and
cosine functions are defined as
follows:
f() sin is the function
whose output is the
y-coordinate.
g() cos is the function
whose output is the
x-coordinate.
y
(x, y) = (cos , sin )
(1, 0)
FIGURE A.24
cos2 sin2 1
Identity:
x
for any angle Figure A.25 shows the four points where the unit circle intersects the axes, the
corresponding angles in radians, and the values of the sine and cosine functions.
π
π =1
π = 0, sin _
= _2 , cos _
2
2
(0, 1)
(-1, 0)
(1, 0)
= 0, cos 0 = 1, sin 0 = 0
= π, cos π = -1, sin π = 0
(0, -1)
_ , cos 3π
_ = -1
_ = 0, sin 3π
= 3π
2
2
2
FIGURE A.25
Any point on the unit circle has multiple corresponding angles with the same initial
5
and terminal sides. For example, an angle of 2 is 1 complete revolution plus a quar-
ter of a revolution 2 2 , resulting in the same point on the unit circle as the
5
angle 2 . Hence cos 2 cos 2 0.
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APPENDIX A
The values of the sine and cosine functions are not obvious for points other than
those shown in Figure A.25. You can use a calculator or computer to find other sine
and cosine values, as illustrated in Example 2. Be sure your technology is set in radian
mode.
EXAMPLE 9
Instructions for using technology
with this example are given in
Section 8.1.1 of the technology
supplement.
Calculating Sine and Cosine Values
9
9
a. Find sin 8 and cos 8 .
b. Interpret your answer to part a in terms of the coordinates of a point on the unit
circle.
c. Find two other angles whose sine and cosine are the same as those in part a.
Solution
9
a. Using a calculator or computer you should find that sin 8 0.38268 and
9
cos 8 0.92388.
y
9π
_
8
x
(-0.92388, -0.38268)
FIGURE A.26
b. The cosine and sine values are the x- and y-coordinates of the point where the ter9
minal side of the angle 8 intersects the unit circle. This angle and the corresponding point are shown in Figure A.26. Note that because the x- and y-coordinates of
the point are negative, both the cosine and the sine of this angle are negative.
c. We seek two other angles corresponding to the point where the terminal side
of the angle shown in Figure A.26 intersects the unit circle. One such angle is
9
obtained by a clockwise rotation from the positive x-axis. This angle is 2 8 7
8 and is expressed as a negative number to denote the direction. Thus the
7
7
9
7
9
angle is 8 , and sin 8 sin 8 and cos 8 cos 8 .
It is also possible to reach the point under consideration by going around the
9
circle 1 full rotation counterclockwise and then an additional 8 radians. This
9
25
25
9
25
9
angle is 2 8 8 . Thus sin 8 sin 8 and cos 8 cos 8 . There are
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Trigonometry Basics
infinitely more positive and negative angles corresponding to the point indicated
in Figure A.26, all with a sine value of approximately 0.38268 and a cosine value
of approximately 0.92388. ●
So far we have referred to f() sin and g() cos as functions, but we have
not verified that this is the case. To verify that f() sin is a function, we must ask,
“Can sin have more than one output value for a particular angle (the input)?”
Because the terminal side of the angle intersects the unit circle at a single point,
there will be only one output f() sin corresponding to each angle . Thus
f() sin is indeed a function. Similar reasoning confirms that g() cos is also
a function.
Be careful not to confuse angle inputs of these functions. Even though angles
13
such as 6 and 6 have the same initial and terminal sides and therefore the same
1
trigonometric function outputs, the two angles are not the same: 6 is 12 of a
13
1
revolution, whereas 6 represents 1 complete revolution plus 12 additional
revolution. For cyclic functions, infinitely many distinct inputs correspond to the
same output.
In a graph of the sine function, the horizontal axis represents the angle measures
and the vertical axis represents the y-coordinates on the unit circle. For the cosine
function, the horizontal axis represents the angle measure and the vertical axis represents the x-coordinate on the unit circle.
Refer to Figure A.25. If we plot the angles shown in that figure and the corresponding y-coordinates, we obtain the graph shown in Figure A.27. We also add the
point (2, 0) corresponding to 1 complete revolution.
We add more points to this graph by calculating some intermediate values for
the sine function. Table A.1 shows selected angles between 0 and 2 (corresponding
to sixteenths of a revolution) and the associated y-coordinates (to five decimal
places) on a unit circle. Figure A.28 shows the points in Table A.1 added to the graph
in Figure A.27.
TABLE A.1
f ()
sin sin 1
8
0.38268
9
8
0.38268
4
0.70711
5
4
0.70711
3
8
0.92388
11
8
0.92388
5
8
0.92388
13
8
0.92388
3
4
0.70711
7
4
0.70711
7
8
0.38268
15
8
0.38268
0
2
_π
2
π
3π
_
2
2π
-1
FIGURE A.27 Points on the graph
of f() sin 0
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APPENDIX A
f ()
f ()
1
1
_π
2
π
3π
_
2
f () = sin _π
2π
2
π
3π
_
2
2π
-1
-1
FIGURE A.28
FIGURE A.29
Because the y-coordinates increase, decrease, and increase again and take on all
real number values between 1 and 1 as we move around the circle, we expect the sine
function graph to increase and decrease in a smooth, continuous manner, taking on
all values between 1 and 1. We therefore connect these points with a smooth curve to
obtain the graph of the sine function shown in Figure A.29.
Once the angle exceeds 2, we begin retracing the unit circle, and the sine function begins repeating itself. Figure A.30 shows a sine graph with more of the repetitions. This graph extends infinitely far in both directions.
f ()
f () = sin 1
-2π
_
-3π
2
-π
- _π2
_π
2
π
3π
_
2
2π
5π
_
2
3π
7π
_
2
4π
-1
FIGURE A.30
The sine function is periodic because it repeats itself every 2 input units. The
period of the sine function is 2. The sine function is also cyclic, because it varies
continuously, alternating between 1 and 1. The portion of the sine function over
one period— that is, the part that keeps repeating itself— is called a cycle of the
function.
Although we have viewed the sine and cosine functions as having angle measure
as input, we are not restricted to this interpretation. In fact, we can consider the sine
and cosine functions as having any real number as the input. Because most applications to which we will apply a sine model have input that is not an angle measure,
we will no longer use to denote the input. Instead we use the notation y sin x
and y cos x. Do not confuse x and y here with points on the unit circle. The variable x is simply the input, which can be interpreted as an angle measured in radians,
and the variable y is the output, which can be interpreted as the x- or y-coordinate on
the unit circle, depending on whether we are considering a cosine function or a sine
function.
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Trigonometry Basics
A.1 Concept Inventory
A19
A.1 Activities
Degree measure of angles
1. Because the number 360 has so many integer divisors, many fractional revolutions have nice expressions when expressed as angles measured in degrees.
Complete the table of fractional rotations or full
rotations and their associated degree measures.
Initial and terminal sides and standard position
of angles
Right-triangle trigonometry: sine, cosine, and
tangent
2. There are many easily expressed fractions of a full
turn, and the measure of these expressions in radians is easy to determine. Complete the table of fractional rotations or full rotations and their associated radian measures.
Unit-circle trigonometry: sine, cosine, and tangent
Radian measure of angles
Angle measure conversion
Sine and cosine functions of a general angle
Table for Activity 1
Rotation (in turns)
1
24
Angle measure
(in degrees)
15°
Rotation (in turns)
Angle measure
(in degrees)
Rotation (in turns)
1
12
1
4
45°
3
8
1
2
135°
180°
7
8
1
Angle measure
(in degrees)
225°
1
3
60°
120°
2
3
5
6
240°
270°
300°
3
60
720°
3780°
Table for Activity 2
Rotation (in turns)
1
24
Angle measure
(in radians)
12
Rotation (in turns)
3
8
Angle measure
(in radians)
Rotation (in turns)
Angle measure
(in radians)
1
12
4
1
2
7
8
1
4
3
5
6
2
3
3
2
5
4
1
3
4
1
3
5
3
60
10.5
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APPENDIX A
3. Convert the following angles in degrees into angles
in radians, and sketch the angles on a unit circle.
a. 110°
b. 700°
c. 90°
d. 0.01°
4. Convert the following angles in radians to angles in
degrees, and sketch the angles on a unit circle.
a. 2
3
b. 8
7
c. 16
d. 30
5. Cycling Superbike II, the $15,000 baby of the U.S.
Cycling Federation (USCF), made its debut in the
1996 Olympics. Superbike II’s front wheel is 23.62
inches in diameter, and its rear wheel is 27.56
inches in diameter. Over a distance of 1 mile, how
many turns does each wheel make?
6. Auto Wheels Use degree measure to write the
angle that the tenths wheel on the odometer turns
when the drive shaft of the automobile in Example
6 completes one turn.
7. Ferris Wheel Consider a Ferris wheel on which
are 30 equally spaced seats.
a. Through what angle does the wheel move
between stops to release passengers in two consecutive seats?
b. What is the radian measurement of this angle?
c. If the Ferris wheel is 100 feet in diameter, what is
the distance traveled by each of the seats as the
operator stops between consecutive seats to
exchange passengers?
8. Auto Wheels Suppose the owner of the automobile in Example 6 replaced the tires with oversized
tires that were 22 inches in diameter.
a. What would be the error created on the odometer readings?
b. If the tires were replaced with tires that have an
18-inch diameter, what would be the error caused
on the odometer?
c. If the regular tires (20-inch diameter) lost 0.001
inch of their diameter through wear, how would
the odometer readings be affected?
9. For finer measurements, the degree is further
divided into 60 parts, and each such part is called a
minute. The symbol used to indicate minutes is the
single prime; for example, 15 minutes is expressed
as 0°15
. An angle consisting of 200 minutes could
also be expressed as 3°20
. For even finer measurements, the minute is further divided into 60 equal
Copyright © Houghton Mifflin Company. All rights reserved.
parts called seconds. The symbol for a second is the
double prime, so an angle of 30 degrees 14 minutes
7 seconds would be written as 30°14
7. You might
try to estimate how small a second is to realize how
much precision is used when some jobs state
acceptable tolerances in terms of a seconds of arc.
Convert into decimal degree equivalents the following angles given in degrees, minutes, and seconds.
(Note: The Saturn missile used in the moon shots
had guidance computers placed on a stable gimbaled platform that had to stay within 4 seconds of
arc for the first 15 minutes of the launch.)
a. 5°30
30
b. 35°12
17
10. Refer to the definitions in Activity 9, and convert
into measures using degrees, minutes, and seconds
the following angles given in decimal degrees.
a. 5.1234°
b. 125.365°
11. Suppose you were to divide a rotation into 1776
pieces. We will call each piece a “patriotic unit” and
will use the abbreviation PU.
a. How many rotations does each of the following
patriotic unit measurements represent?
i. 1776 PU
ii. 1332 PU
iii. 888 PU
iv. 444 PU
v. 222 PU
vi. 111 PU
b. Convert each of the patriotic unit measurements
in part a to radians.
c. Find the value of each of the following:
i. cos(1776 PU)
ii. sin(1776 PU)
iii. cos(666 PU)
iv. sin(666 PU)
12. Clock Creating different angle measurement systems such as degrees and grads for partial turns and
calculating their conversion factors is like making a
25-hour clock. Mr. Morton Rachofsky has built a
25-hour clock for the Circadian Clock Company.
This clock divides the 86,400 seconds in the standard
day into 25 equal-length periods called “hours.”
Noon on the clock is the same as noon on our regular time scale, but the other hour marks are different.
Scientists conducting experiments in the 1930s
observed people in caves where they could not see
the sun. These people developed activity cycles that
lasted 25 hours. Because we cannot change the solar
day, Mr. Rachofsky said, “Why not change the clock?”
(Source: New York Times. October 27, 1996, p. 47.)
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Trigonometry Basics
a. How long are Mr. Rachofsky “hours” in our regular minutes?
b. What time on Mr. Rachofsky’s clock is the regular clock time 3:00 P.M.?
c. What is the regular clock time when it is 6:00
P.M. on Mr. Rachofsky’s clock?
For Activities 13 through 18, identify the trigonometric
function you would use to find the length of the indicated side. Assume that the triangle given is a right triangle with one of the other two angles identified as
angle .
13. Given the length of the hypotenuse, find the length
of the leg adjacent to angle .
14. Given the length of the leg opposite angle , find
the length of the leg adjacent to angle .
15. Find the length of the leg opposite angle , given
the length of the leg adjacent to angle .
16. Find the length of the hypotenuse, given the length
of the leg opposite angle .
17. Given the length of the leg adjacent to angle , find
the length of the hypotenuse.
A21
22. One angle of the triangle is 30.75°. The leg opposite
this angle is 5 inches long.
a. Find the length of the leg adjacent to the given
angle.
b. Find the length of the hypotenuse.
23. One angle of the triangle is 85.4°. The leg adjacent
to this angle is 1.5 miles long.
a. Find the length of the hypotenuse.
b. Find the length of the leg opposite the given
angle.
24. One angle of the triangle is 10.2°. The hypotenuse
is 3 centimeters long.
a. Find the length of the leg adjacent to the given
angle.
b. Find the length of the leg opposite the given
angle.
25. Roof A gable is the triangular segment of a wall
created by the roof. The pitch of a gable is its height
divided by its width. See figure. (That is, pitch is
equal to half the slope of the roof.) Consider a gable
40 feet wide feet with an angle of 130° at its peak.
18. Given the length of the hypotenuse, find the length
of the leg opposite angle .
For Activities 19 through 24, solve for the length of the
indicated side. Assume that the triangle given is a right
triangle.
19. One angle of the triangle is 20°. The leg opposite
this angle is 5 inches long.
a. Find the length of the leg adjacent to the given
angle.
b. Find the length of the hypotenuse.
20. One angle of the triangle is 78°. The leg adjacent to
this angle is 1 meter long.
a. Find the length of the hypotenuse.
b. Find the length of the leg opposite the given
angle.
21. One angle of the triangle is 15.2°. The hypotenuse
is 12 centimeters long.
a. Find the length of the leg adjacent to the given
angle.
b. Find the length of the leg opposite the given
angle.
Height
Width
a. How tall is the attic at its center?
b. What is the pitch of the gable?
c. How much board would be needed to cover the
roof (not including the overhanging portion of
the roof) if the house is 40 feet long?
26. Nuts Find the diameter of the smallest iron rod
from which a hexagonal nut with side 4mm can be
cut. (Hint: The angle between two adjacent sides of
the nut is 120°.)
27. Stairway A stairway is to be constructed on a hill
with a 34° incline.
a. If each step is to have a 7-inch rise, what must be
its tread (horizontal depth)?
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APPENDIX A
b. How many 7-inch steps will be needed if the
length of the hill (measured up the slope) is 4
feet 2 inches?
28. Navigation A boat that is sailing S 47°E is sailing
on a trajectory that is along an angle 47° east of true
south. A ship sails 12.8 nautical miles S 47°E from
its starting position.
a. How far south has the ship sailed?
b. How far east has it sailed?
29. Navigation Air Force pilots mark their bearing as
the clockwise angle measured from the north. For
example, a bearing of 90° is east and a bearing of
180° is south. There is a landing strip 5 miles away
from a plane at bearing 332°.
a. What bearing is west?
b. In what direction must the plane fly to reach the
landing strip?
c. If the plane were to fly directly north or south
and then east or west to reach the landing strip,
how far in each direction would the plane have
to fly?
30. Because the sum of the three angles of any triangle
is 180°, a right triangle with one 45° angle must
contain another 45° angle. As shown, this type of
triangle is formed by two sides and a diagonal of a
square. Thus the two legs of the triangle are of
equal length. By the Pythagorean Theorem, the
hypotenuse must have length
s2
s2
2s2
32. 160°
33. 415°
34. 920°
35. 310°
36. 489°
37. 945°
38. 280°
39. 37°
40. Calculate sin 180° and cos 180° and explain your
answers in terms of the unit circle. Do the same
thing for sin 90° and cos 90°.
41. a. Use the graph to estimate sin 4, cos 4, sin 8 ,
and cos 8 .
y
1
−1
=4
=
–
8
1 x
−1
2s
b. Mark the estimated locations of the angles
45°
√2 s
For Activities 32 through 39, sketch the given angle on
the unit circle, and draw the appropriate right triangle
corresponding to this angle. Calculate the sine and
cosine of the angle, and indicate these values appropriately on the sketch of the triangle.
144
s
1 and 16 on the figure, and use the
grid to estimate the values of sin(1), cos(1), sin
144
144
16 , and cos 16 .
45°
s
Using the lengths s and 2s, determine the exact
values of the following trigonometric ratios.
a. sin 45°
b. cos 45°
c. tan 45°
31. Use the values for sin 45° and cos 45° to determine
the sine, cosine, and tangent values of each of the
following angles.
a. 315°
b. 135°
c. 225°
Copyright © Houghton Mifflin Company. All rights reserved.
42. a. Use the graph to estimate sin 2.5, cos 2.5,
15
15
sin 16 , and cos 16 .
b. Mark the estimated locations of the angles
1
5
2 and 12 on the figure, and use the
1
1
grid to estimate the values of sin 2 , cos 2 ,
5
5
sin 12 , and cos 12 .
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Trigonometry Basics
y
49. a. Without using technology, indicate whether each
of the following values is positive, negative, or zero
by using its interpretation as the y-coordinate of
a point on the unit circle.
1
−1
= 2.5
1
= –15
16
Trig value
x
sin 0
sin 4
3
sin 2
sin 4
sin Sign
5
Trig value
7
3
sin 4
sin 2
sin 2 sin 4
Sign
−1
2
b. Use your calculator or computer to complete the
following table. Compare the signs of the values
to your results from part a.
2
43. a. Find the values of sin 3 and cos 3 .
b. Interpret each answer to part a in terms of the
coordinates of a point on the unit circle.
c. Give two other angles whose sine and cosine val2
ues are the same as those for 3 .
Trig value
sin 0
sin 4
sin 2
3
sin 4
sin Decimal value
5
Trig value
sin 4
3
sin 2
7
sin 4
sin 2
Decimal value
44. a. Find the values of sin 15 and cos 15.
b. Interpret each answer to part a in terms of the
coordinates of a point on the unit circle.
c. Give two other angles whose sine and cosine values are the same as those for 15.
3
3
45. a. Find the values of sin 2 and cos 2 .
b. Interpret the answers to part a in terms of the
unit circle.
c. Give three other angles whose sine and cosine
3
values are the same as those for 2 .
4
4
46. a. Find the values of sin 9 and cos 9 .
b. Interpret the answers to part a in terms of the
unit circle.
c. Give three other angles whose sine and cosine
4
values are the same as those for 9 .
47. Is it possible for the trigonometric functions of an
angle to be cos 0.5 and sin 0.5? Explain.
c. Plot the values you obtained in part b as a function of the angle. How is this plot related to the
graph of the function f(x) sin x?
50. a. Without using technology, indicate whether
each of the following values is positive, negative,
or zero by using its interpretation as the xcoordinate of a point on the unit circle.
Trig value
cos 0
cos 4
3
cos 2
cos 4
cos Sign
Trig value
5
7
3
cos 4
cos 2
cos 4
cos 2 Sign
b. Use technology to complete the following table.
Compare the signs of the values to your results
from part a.
Trig value
cos 0
cos 4
cos 2
3
cos 4
cos Decimal value
48. Is it possible for the trigonometric functions of an
angle to be cos 0.35 and sin 0.82?
Explain.
Trig value
5
cos 4
3
cos 2
7
cos 4
cos 2 Decimal value
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A25
Answers to Appendix A
Answers to Appendix A
TRIGONOMETRY BASICS
1 1 5 3
21
1. Rotation (in turns): 8 , 6 , 8 , 4 , 2, 2
Angle measure (in degrees): 30, 90, 315, 360, 1080, 21,600
11
11π
___
18
3. a. 18 radians
b.
25. a.
b.
c.
27. a.
b.
29. a.
b.
c.
18.643 miles
9.326 feet
0.233
1765.4 square feet
10.38 inches
Four steps
Bearing 270°
Northwest
4.415 miles north and 2.347 miles west
31.
sin cos 2
2
2
2
2
2
2
2
b. 135°
2
2
2
2
1
c. 225°
2
2
2
2
1
45°
b.
35
9 radians
a. 315°
35π
___
9
π
_
2
c. 2 radians
0.000056
d. 0.000056 radian
3
1
1
iv. 4 rotation v. 8 rotation
b. i. 2 radians ii. 1.5 radians
13.
15.
17.
19.
21.
23.
iv. 2 radians
v. 4 radians
c. i. 1
ii. 0
iii. 0.707
cos tan cos a. 13.737 inches
b. 14.619 inches
a. 11.580 centimeters
b. 3.146 centimeters
a. 18.703 miles
1
sin cos 33.
415°
0.819
0.574
35.
310°
0.766
0.643
37.
945°
0.707
0.707
39.
37°
0.602
0.799
y
33.
ii. 4 rotation
1
5. The front wheel makes approximately 853.86 turns in 1
mile; the rear wheel makes approximately 731.79 turns in
1 mile.
7. a. 12°
b. 15 radian
c. Approximately 10.47 feet
9. a. Approximately 5.508°
b. Approximately 35.2047°
11. a. i. 1 rotation
tan 0.819
415°
x
0.574
1
iii. 2 rotation
1
vi. 16 rotation
iii. radians
vi. 8 radians
iv. 0.707
y
35.
310°
0.643
x
-0.766
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APPENDIX A
37.
y
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Page 26
y
1
0.707
=
-945°
x
-0.707
144
16
–1
1
x
= –1
–1
y
39.
2
2
43. a. sin 3 0.866, cos 3 0.500
b. The point on the unit circle corresponding to an angle
2
0.799
-37°
of 3 is approximately (0.5, 0.866).
4
8
c. Two possible answers are 3 and 3 .
x
-0.602
3
3
45. a. sin 2 1, cos 2 0
b. The point on the unit circle corresponding to an angle
3
41. a. sin 4 0.8, cos 4 0.7, sin 8 0.4,
cos 8 0.9 (Answers may vary.)
144
b. sin (1) 0.8, cos(1) 0.5, sin 16 0,
144
cos 16 1 (Answers may vary.)
Copyright © Houghton Mifflin Company. All rights reserved.
of 2 is (0, 1).
11
7
c. Possible answers include 2 , 2 , and 2 .
47. It is not possible to have an angle such that cos 0.5
and sin 0.5 because cos and sin are x- and ycoordinates on the unit circle and must satisfy the equation cos2 sin2 1.
49. a. Zero, positive, positive, positive, zero
Negative, negative, negative, zero
b. 0, 0.707, 1, 0.707, 0
0.707, 1, 0.707, 0