Download 14.1 Angles and Their Measures 14.2 Trigonometric

he 1958 musical Merry Andrew starred Danny Kaye as
Andrew Larabee, a teacher with a flair for using unconventional methods in his classes. He uses a musical number
to teach the Pythagorean theorem, singing and dancing to
“The Square of the Hypotenuse”:
T
. . . Parallel lines don’t connect, which is just about what you
might expect.Though scientific laws may change and decimals
can be moved, the following is constant, and has yet to be disproved: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two
adjacent sides.
The Pythagorean theorem, introduced in Chapter 9, is
used extensively in the study of trigonometry, the foundations
of which go back at least 3000 years.The word trigonometry
comes from the Greek words for triangle (trigon) and measurement (metry). Today trigonometry is used in electronics,
surveying, and other engineering areas, and is necessary for
further courses in mathematics, such as calculus.
14.1 Angles and Their Measures
14.2 Trigonometric Functions of Angles
14.3 Trigonometric Identities
14.4
Right Triangles and Function Values
14.5
Applications of Right Triangles
14.6 The Laws of Sines and Cosines
Extension Area Formulas for Triangles
Collaborative Investigation Making a Point
about Trigonometric Function Values
Chapter 14 Test
749
750
CHAPTER 14
Trigonometry
14.1 ANGLES AND THEIR MEASURES
Basic Terminology • Degree Measure • Angles in a Coordinate System
A
A
B
B
A
Line AB
Segment AB
B
Ray AB
Figure 1
Terminal side
Vertex A
Initial side
Basic Terminology
A line may be drawn through the two distinct points A and B. This line is called
line AB. The portion of the line between A and B, including points A and B
themselves, is segment AB. The portion of the line AB that starts at A and continues through B, and on past B, is called ray AB. Point A is the endpoint of the
ray. See Figure 1.
An angle is formed by rotating a ray around its endpoint. The ray in its initial
position is called the initial side of the angle, while the ray in its location after the
rotation is the terminal side of the angle. The endpoint of the ray is the vertex of
the angle. See Figure 2.
If the rotation of the terminal side is counterclockwise, the angle measure
is positive. If the rotation is clockwise, the angle measure is negative. See
Figure 3.
Figure 2
A
C
B
Positive angle
Negative angle
Figure 3
An angle can be named by using the name of its vertex. For example, the angle
on the right in Figure 3 can be called angle C. Alternatively, an angle can be named
using three letters, with the vertex letter in the middle. Thus, the angle on the right
also could be named angle ACB or angle BCA.
Degree Measure
A complete rotation of a ray gives an
angle whose measure is 360°.
Figure 4
There are two systems in common use for measuring the sizes of angles. The most
common unit of measure is the degree. (The other common unit of measure is called
the radian.) Degree measure was developed by the Babylonians 4000 years ago. To
use degree measure we assign 360 degrees to a complete rotation of a ray. In Figure 4,
notice that the terminal side of the angle corresponds to its initial side when it
makes a complete rotation.
1
One degree, written 1°, represents 360
of a rotation. Therefore, 90° represents
90
1
180
1
=
of
a
complete
rotation,
and
180°
represents
360
4
360 = 2 of a complete rotation.
Angles of measure 5°, 90°, and 180° are shown in Figure 5.
90°
5° angle
Figure 5
180°
14.1
”Trigonometry, perhaps more than any
other branch of mathematics, developed
as the result of a continual and fertile
interplay of supply and demand: the supply
of applicable mathematical theories and
techniques available at any given time and
the demands of a single applied science,
astronomy. So intimate was the relation
that not until the thirteenth century was
it useful to regard the two subjects as
separate entities.” (From “The History of
Trigonometry” by Edward S. Kennedy, in
Historical Topics for the Mathematics
Classroom, the Thirty-first Yearbook of
N.C.T.M., 1969.)
Angles and Their Measures
751
Special angles are named as shown in the following chart.
Name
Acute angle
Angle Measure
Example(s)
Between 0° and 90°
60°
Right angle
82°
Exactly 90°
90°
Obtuse angle
Between 90° and 180°
138°
97°
Straight angle
Exactly 180°
180°
If the sum of the measures of two angles is 90°, the angles are called
complementary. Two angles with measures whose sum is 180° are supplementary.
EXAMPLE 1
Finding Complement and Supplement
Give the complement and the supplement of 50°.
SOLUTION
The complement of 50° is 90° - 50° = 40°.
The supplement of 50° is 180° - 50° = 130°.
Do not confuse an angle with its measure. The angle itself consists of the vertex
together with the initial and terminal sides. The measure of the angle is the size of the
rotation angle from the initial to the terminal side (commonly expressed in
degrees).
For example, if angle A has a 35° rotation angle, we say that m1angle A2 is 35°,
where m1angle A2 is read “the measure of angle A.”We abbreviate m1angle A2 = 35°
as simply angle A = 35°, or just A = 35°.
Traditionally, portions of a degree have been measured with minutes and sec1
onds. One minute, written 1 œ , is 60
of a degree.
1¿ ⴝ
1 °
or 60¿ ⴝ 1°
60
1
One second, 1 fl , is 60
of a minute.
1– ⴝ
1 ¿
1 °
ⴝ
or 60– ⴝ 1¿
60
3600
The measure 12° 42¿ 38– represents 12 degrees, 42 minutes, 38 seconds.
752
CHAPTER 14
Trigonometry
EXAMPLE 2
Calculating with Degree Measure
Perform each calculation.
(a) 51° 29¿ + 32° 46¿
51°29'32°46'
SOLUTION
(a) Add the degrees and the minutes separately.
䉴 DMS
84°15'0''
90°0'73°12'
(b) 90° - 73° 12¿
51° 29¿
+ 32° 46¿
83° 75¿
䉴 DMS
16°48'0''
The calculations explained in Example 2
can be done with a graphing calculator
capable of working with degrees, minutes,
and seconds.
(b)
90°
- 73° 12¿
Simplify.
83°
+ 1° 15¿
84° 15¿
Write 90° as 89° 60 ¿.
75 ¿ = 60 ¿ + 15 ¿ = 1° 15 ¿
89° 60¿
- 73° 12¿
16° 48¿
Angles can be measured in decimal degrees. For example, 12.4238° represents
12.4238° = 12
EXAMPLE 3
4238 °
.
10,000
Converting between Decimal Degrees and Degrees,
Minutes, Seconds
(a) Convert 74° 8¿ 14– to decimal degrees. Round to the nearest thousandth of a
degree.
(b) Convert 34.817° to degrees, minutes and seconds. Round to the nearest second.
SOLUTION
74°8'14''
74.13722222
8 °
14 °
+
60
3600
L 74° + 0.1333° + 0.0039°
= 74.137°
(a) 74° 8¿ 14– = 74° +
74°8'14''
74.137
34.817 䉴 DMS
34°49'1.2''
The conversions in Example 3 can be
done on some graphing calculators. The
second displayed result was obtained by
setting the calculator to show only three
places after the decimal point.
(b) 34.817° =
=
=
=
=
=
L
34° + 0.817°
34° + 10.8172(60 ¿2
34° + 49.02¿
34° + 49¿ + 0.02¿
34° + 49¿ + 10.022160–2
34° + 49¿ + 1.2–
34° 49¿ 1–
1¿ =
1 °
60 and
1– =
1 °
3600
Add. Round to three decimal places.
1° = 60 ¿
1 ¿ = 60–
Angles in a Coordinate System
An angle u (the Greek letter theta)* is in standard position if its vertex is at the
origin of a rectangular coordinate system and its initial side lies along the positive x-axis. The two angles shown in Figures 6(a) and (b) on the next page are in
standard position. An angle in standard position is said to lie in the quadrant
in which its terminal side lies. For example, an acute angle is in quadrant I and
an obtuse angle is in quadrant II.
*The letters of the Greek alphabet are identified in a margin note on page 758.
14.1
Angles and Their Measures
753
Figure 6(c) shows ranges of angle measures for each quadrant when 0° 6 u 6
360°. Angles in standard position having their terminal sides along the x-axis or
y-axis, such as angles with measures
90°,
180°,
270°, and so on,
are called quadrantal angles.
90°
y
y
QI
Q II
Q II
90° < ␪ < 180°
Terminal side
␪
Vertex 0
x
0
Initial side
0°
360°
180°
␪
x
QI
0° < ␪ < 90°
Q III
Q IV
180° < ␪ < 270° 270° < ␪ < 360°
270°
(a)
(b)
(c)
Figure 6
A complete rotation of a ray results in an angle of measure 360°. If the rotation
is continued, angles of measure larger than 360° can be produced. The angles in
Figure 7(a) have measures 60° and 420°. These two angles have the same initial side
and the same terminal side, but different amounts of rotation. Angles that have the
same initial side and the same terminal side are called coterminal angles. As shown
in Figure 7(b), angles with measures 110° and 830° are coterminal.
y
y
60°°
x
0
(a)
x
0
Coterminal
angles
y
830°
110°°
420°°
Coterminal
angles
(b)
Figure 7
188°
908°
x
0
EXAMPLE 4
Finding Measures of Coterminal Angles
Find the angle of least possible positive measure coterminal with each angle.
(a) 908°
Figure 8
(b) - 75°
SOLUTION
(a) Add or subtract 360° from 908° as many times as needed to get an angle with
measure greater than 0° but less than 360°. Because
y
908° ⴚ 2
0
285°
#
360° = 908° ⴚ 720° = 188°,
x
–75°
an angle of 188° is coterminal with an angle of 908°. See Figure 8.
(b) See Figure 9. Use a rotation of
Figure 9
360° + 1- 75°2 = 285°.
754
CHAPTER 14
Trigonometry
Sometimes it is necessary to find an expression that will generate all angles
coterminal with a given angle. For example, because any angle coterminal with 60°
can be obtained by adding an appropriate integer multiple of 360° to 60°, we can let
n represent any integer, and the expression
60° ⴙ n
#
360°
will represent all such coterminal angles. Table 1 shows a few possibilities.
Table 1
Value of n
Angle Coterminal with 60°
2
60° + 2
#
360° = 780°
1
60° + 1
#
360° = 420°
0
60° + 0
#
360° = 60° (the angle itself)
-1
60° + 1- 12
#
360° = ⴚ300°
14.1 EXERCISES
Give (a) the complement and (b) the supplement of each angle.
1. 30°
2. 60°
3. 45°
4. 55°
5. 89°
6. 2°
7. If an angle measures x degrees, how can we represent
its complement?
8. If an angle measures x degrees, how can we represent
its supplement?
Perform each calculation.
9. 62° 18 ¿ + 21° 41 ¿
10. 75° 15¿ + 83° 32 ¿
11. 71° 58 ¿ + 47° 29 ¿
12. 90° - 73° 48¿
13. 90° - 51° 28 ¿
14. 180° - 124° 51¿
15. 90° - 72° 58 ¿ 11–
16. 90° - 36° 18¿ 47–
Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree.
Convert each angle measure to degrees, minutes, and seconds. Use a calculator, and round to the nearest second.
23. 31.4296°
24. 59.0854°
25. 89.9004°
26. 102.3771°
27. 178.5994°
28. 122.6853°
Find the angle of least positive measure coterminal with each
angle.
29. - 40°
30. - 98°
31. - 125°
32. - 203°
33. 539°
34. 699°
35. 850°
36. 1000°
Give an expression that generates all angles coterminal with
the given angle. Let n represent any integer.
37. 30°
38. 45°
39. 60°
40. 90°
Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of
two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each
angle.
17. 20° 54 ¿
18. 38° 42¿
19. 91° 35 ¿ 54–
20. 34° 51¿ 35–
41. 75°
42. 89°
43. 174°
44. 234°
21. 274° 18 ¿ 59–
22. 165° 51¿ 9–
45. 300°
46. 512°
47. - 61°
48. - 159°
14.2
Trigonometric Functions of Angles
755
14.2 TRIGONOMETRIC FUNCTIONS OF ANGLES
Trigonometric Functions • Undefined Function Values
y
Trigonometric Functions
P(x, y)
⎩
⎪
⎧
⎪
⎪
y⎨
⎪
⎪
⎩
⎪
⎪
⎨
r
⎪
⎪
⎪
θ
⎪
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎧
Q
x
x
O
Figure 10
The study of trigonometry covers the six trigonometric functions defined in this section. To define these six basic functions, start with an angle u in standard position.
Choose any point P having coordinates 1x, y2 on the terminal side of angle u. (The
point P must not be the vertex of the angle.) See Figure 10.
A perpendicular from P to the x-axis at point Q determines a triangle having
vertices at O, P, and Q. The distance r from P1x, y2 to the origin, 10, 02, can be found
from the distance formula.
r = 21x - 022 + 1y - 022
Distance formula
r = 2x 2 + y 2
Notice that r>0, because this distance is never negative.
The six trigonometric functions of angle u are called sine, cosine, tangent,
cotangent, secant, and cosecant. In the following definitions, we use the customary
abbreviations for the names of these functions: sin, cos, tan, cot, sec, csc.
Definitions of the Trigonometric Functions
Let 1x, y2 be a point other than the origin on the terminal side of an angle u in
standard position. The distance from the point to the origin is r = 2x 2 + y 2.
The six trigonometric functions of u are defined as follows.
y
y
x
sin U ⴝ
cos U ⴝ
tan U ⴝ
1x ⴝ 02
r
r
x
csc U ⴝ
r
y
1y ⴝ 02
sec U ⴝ
r
x
1x ⴝ 02
cot U ⴝ
x
y
1y ⴝ 02
Although Figure 10 shows a second quadrant angle, these definitions apply to
any angle u. Due to the restrictions on the denominators in the definitions of tangent, cotangent, secant, and cosecant, quadrantal angles will have some undefined
function values.
y
EXAMPLE 1
(8, 15)
17
15
The terminal side of an angle u in standard position passes through the point 18, 152.
Find the values of the six trigonometric functions of angle u.
x= 8
y = 15
r = 17
θ
0
8
Figure 11
Finding Function Values of an Angle
x
SOLUTION
Figure 11 shows angle u and the triangle formed by dropping a perpendicular from
the point 18, 152 to the x-axis. The point 18, 152 is 8 units to the right of the y-axis
and 15 units above the x-axis, so that x = 8 and y = 15.
r = 2x 2 + y 2
r = 282 + 15 2
Let x 8 and y 15.
r = 264 + 225
Square 8 and 15.
r = 2289
Add.
r = 17
Find the square root.
756
CHAPTER 14
Trigonometry
The values of the six trigonometric functions of angle u can now be found with
the definitions given in the box, where x = 8, y = 15, and r = 17.
y
15
=
r
17
r
17
csc u = =
y
15
sin u =
EXAMPLE 2
x
8
=
r
17
r
17
sec u = =
x
8
cos u =
y
15
=
x
8
x
8
cot u = =
y
15
tan u =
Finding Function Values of an Angle
The terminal side of an angle u in standard position passes through the point
1- 3, - 42. Find the values of the six trigonometric functions of u.
SOLUTION
As shown in Figure 12, x = - 3 and y = - 4. Find the value of r.
y
θ
–3
sin u =
(–3, –4)
Figure 12
y
OP = r (x′, y′)
P′
OP′ = r′
P
θ
O
Q
Q′
x
Remember that r 7 O.
Then use the definitions of the trigonometric functions.
-4
4
= 5
5
5
5
csc u =
= -4
4
5
(x, y)
r = 225
r = 5
x
0
–4
r = 21- 322 + 1- 422
x = –3
y = –4
r = 5
-3
3
= 5
5
5
5
sec u =
= -3
3
cos u =
-4
4
=
-3
3
-3
3
cot u =
=
-4
4
tan u =
The six trigonometric functions can be found from any point on the terminal
side of the angle other than the origin. To see why any point may be used, refer to
Figure 13, which shows an angle u and two distinct points on its terminal side. Point
P has coordinates 1x, y2 and point P¿ (read “P-prime”) has coordinates 1x¿, y¿2. Let
r be the length of the hypotenuse of triangle OPQ, and let r¿ be the length of the
hypotenuse of triangle OP¿Q¿. Because corresponding sides of similar triangles are
in proportion,
y
y¿
,
=
r
r¿
Figure 13
y
and thus sin u = r is the same no matter which point is used to find it. Similar results
hold for the other five functions.
Undefined Function Values
If the terminal side of an angle in standard position lies along the y-axis, any point on
this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the
x-axis has y-coordinate 0 for any point on the terminal side. Because the values of x and
y appear in the denominators of some of the trigonometric functions, and because a
fraction is undefined if its denominator is 0, some of the trigonometric function values of quadrantal angles will be undefined.
EXAMPLE 3
Finding Function Values and Undefined Function Values
Find values of the trigonometric functions for each angle. Identify any that are
undefined.
(a) an angle of 90°
(b) an angle in standard position with terminal side through 1- 3, 02
14.2
(0, 1)
90°
x
(a)
y
1
= 1
1
1
csc 90° =
= 1
1
0
= 0
1
1
sec 90° = 1undefined2
0
sin 90° =
1
1undefined2
0
0
cot 90° = = 0
1
cos 90° =
tan 90° =
(b) Figure 14(b) shows the angle. Here, x = - 3, y = 0, and r = 3, so the trigonometric functions have the following values.
θ
x
0
(–3, 0)
757
SOLUTION
(a) First, select any point on the terminal side of a 90° angle. We select the point
10, 12, as shown in Figure 14(a). Here x = 0 and y = 1. Verify that r = 1. Then,
use the definitions of the trigonometric functions.
y
0
Trigonometric Functions of Angles
(b)
Figure 14
0
= 0
3
3
csc u = 1undefined2
0
sin u =
-3
= -1
3
3
sec u =
= -1
-3
cos u =
0
= 0
-3
-3
cot u =
1undefined2
0
tan u =
Undefined Function Values
If the terminal side of a quadrantal angle lies along the y-axis, the tangent and
secant functions are undefined. If it lies along the x-axis, the cotangent and
cosecant functions are undefined.
Because the most commonly used quadrantal angles are 0°, 90°, 180°, 270° and
360°, the values of the functions of these angles are summarized in Table 2.
Table 2
U
Function Values of Quadrantal Angles
sin U
cos U
tan U
cot U
sec U
csc U
0°
0
1
0
Undefined
1
Undefined
90°
1
0
Undefined
0
Undefined
1
180°
0
-1
0
Undefined
-1
Undefined
270°
-1
0
Undefined
0
Undefined
-1
360°
0
1
0
Undefined
1
Undefined
14.2 EXERCISES
In Exercises 1– 4, sketch an angle u in standard position such
that u has the least possible positive measure, and the given
point is on the terminal side of u.
1. 1 - 3, 42
2. 1- 4, - 32
3. 15, - 122
4. 1- 12, - 52
Find the values of the trigonometric functions for the angles in
standard position having the following points on their terminal
sides. Identify any that are undefined. Rationalize denominators when applicable.
5. 1 - 3, 42
6. 1- 4, - 32
7. 10, 22
8. 1- 4, 02
9. 11, 232
10. 1- 223, - 22
11. 13, 52
12. 1- 2, 72
13. 1- 8, 02
758
CHAPTER 14
Trigonometry
14. 10, 92
23. IV,
15. For any nonquadrantal angle u, sin u and csc u will have
the same sign. Explain why this is so.
x
r
24. IV,
y
r
25. IV,
y
x
26. IV,
Use the appropriate definition to determine each function
value. If it is undefined, say so.
16. If cot u is undefined, what is the value of tan u?
17. How is the value of r interpreted geometrically in the
definitions of the sine, cosine, secant, and cosecant
functions?
27. cos 90°
28. sin 90°
29. tan 90°
30. cot 90°
31. sec 90°
32. csc 90°
18. If the terminal side of an angle u is in quadrant III, what
is the sign of each of the trigonometric function values
of u?
33. sin 180°
34. sin 270°
35. tan 180°
36. cot 270°
Suppose that the point 1x, y2 is in the indicated quadrant.
Decide whether the given ratio is positive or negative.
(Hint: It may be helpful to draw a sketch.)
37. sin1- 270°2
38. cos1- 270°2
39. tan 0°
40. sec1- 180°2
y
19. II,
r
41. cos 180°
42. cot 0°
20. II,
x
r
y
21. III,
r
22. III,
x
y
x
r
14.3 TRIGONOMETRIC IDENTITIES
Reciprocal Identities • Signs of Function Values In Quadrants • Pythagorean Identities
• Quotient Identities
Reciprocal Identities
The Greek Alphabet
a
alpha
b
beta
g
gamma
d
delta
P
epsilon
z
zeta
h
eta
u
theta
i
iota
k
kappa
l
lambda
m
mu
n
nu
j
xi
o
omicron
p
pi
r
rho
s
sigma
t
tau
y
upsilon
f
phi
x
chi
c
psi
v
omega
The definitions of the trigonometric functions on page 755 were written so that functions directly above and below one another are reciprocals of each other. Because
y
sin u = r and csc u = yr ,
sin u =
1
csc u
and csc u =
1
.
sin u
Also, cos u and sec u are reciprocals, as are tan u and cot u. The reciprocal identities
hold for any angle u that does not lead to a zero denominator.
Reciprocal Identities
sin U ⴝ
1
csc U
cos U ⴝ
1
sec U
tan U ⴝ
1
cot U
csc U ⴝ
1
sin U
sec U ⴝ
1
cos U
cot U ⴝ
1
tan U
Identities are equations that are true for all meaningful values of the variable.
When studying identities, be aware that various forms exist. For example,
sin U ⴝ
1
csc U
can also be written
csc U ⴝ
1
sin U
and
You should become familiar with all forms of these identities.
1sin U21csc U2 ⴝ 1.