Download Additional Topics in Trigonometry Outline

Additional
Topics in
Trigonometry
Outline
Application
7-1 Law of Sines
7-2 Law of Cosines
7-3 Geometric Vectors
7-4 Algebraic Vectors
7-5 Polar Coordinates and Graphs
7-6 Complex Numbers in
Rectangular and Polar Forms
7-7 De Moivre’s Theorem
Chapter 7 Group Activity: Conic
Sections and Planetary Orbits
Chapter 7 Review
Two fire towers are located 11.5 miles apart on a
straight road. Observers at each tower spot a fire
and report its location in terms of the angles in the
figure. A command post is to be set up on the road
as close to the fire as possible. Find the distance (to
the nearest tenth of a mile) from the command
post to tower A and from the command post to the
fire.
n this chapter a number of additional topics involving trigonometry
are considered. First, we return to the problem of solving triangles—
not just right triangles, but any triangle. Then some of these ideas
are used to develop the important concept of vector. With our knowledge of trigonometry, we introduce the polar coordinate system,
probably the most important coordinate system after the rectangular
coordinate system. After considering polar equations and their graphs,
we represent complex numbers in polar form. Once a complex number is in polar form, it will be possible to find nth powers and nth
roots of the number using an ingenious theorem due to De Moivre.
I
Preparing for This Chapter
Before getting started on this chapter, review the following concepts:
Rational Exponents (Appendix A, Section 6)
Radicals (Appendix A, Section 7)
Complex Numbers (Chapter 2, Section 4)
Inverse Functions (Chapter 4, Section 2)
Solving Right Triangles (Chapter 5, Section 5)
Difference Identities (Chapter 6, Section 2)
Significant Digits (Appendix C)
Section 7-1 Law of Sines
Law of Sines Derivation
Solving the ASA and AAS Cases
Solving the SSA Case—Including the Ambiguous Case
The law of sines (developed in this section) and the law of cosines (developed in
the next section) play fundamental roles in solving oblique triangles—triangles
without a right angle. Every oblique triangle is either acute, all angles between
0° and 90°, or obtuse, one angle between 90° and 180°. Figure 1 illustrates both
types of triangles.
FIGURE 1
Oblique triangles.
504
7-1 Law of Sines
505
Note how the sides and angles of the oblique triangles in Figure 1 have been
labeled: Side a is opposite angle ␣, side b is opposite angle ␤, and side c is opposite angle ␥. Also note that the largest side of a triangle is opposite the largest
angle. Given any three of the six quantities indicated in Figure 1, we are interested in finding the remaining three, if they exist. This process is called solving
the triangle.
Before proceeding with specific examples, it is important to recall the rules in
Table 1 regarding accuracy of angle and side measure. Table 1 is also repeated
inside the back cover of the text for easy reference.
T A B L E
1 Triangles and Significant Digits
Angle to Nearest
Significant Digits for Side Measure
1°
10⬘ or 0.1°
1⬘ or 0.01°
10⬙ or 0.001°
2
3
4
5
CALCULATOR CALCULATIONS
When solving for a particular side or angle, carry out all operations
within the calculator and then round to the appropriate number of significant digits (as specified in Table 1) at the end of the calculation. Your
answers may still differ slightly from those in the book, depending on the
order in which you solve for the sides and angles.
Law of Sines Derivation
FIGURE 2
The law of sines is relatively easy to prove using the right triangle properties studied in Section 5-5. We will also use the fact that
sin (180° ⫺ x) ⫽ sin x
which is readily obtained using a different identity (a good exercise for you).
Referring to the triangles in Figure 2, we proceed as follows: For each triangle,
sin ␣ ⫽
h
b
and
sin ␤ ⫽
h
a
Solving each equation for h, we obtain
h ⫽ b sin ␣
and
h ⫽ a sin ␤
Thus,
b sin ␣ ⫽ a sin ␤
sin ␣ sin ␤
⫽
a
b
(1)
506
7 ADDITIONAL TOPICS IN TRIGONOMETRY
Similarly, for each triangle in Figure 2,
sin ␣ ⫽
m
c
and
sin ␥ ⫽ sin (180° ⫺ ␥) ⫽
m
a
Solving each equation for m, we obtain
m ⫽ c sin ␣
m ⫽ a sin ␥
and
Thus,
c sin ␣ ⫽ a sin ␥
sin ␣ sin ␥
⫽
a
c
(2)
If we combine equations (1) and (2), we obtain the law of sines.
LAW OF SINES
1
sin ␣ sin ␤ sin ␥
⫽
⫽
a
b
c
In words, the ratio of the sine of an angle to its opposite side is the same
as the ratio of the sine of either of the other angles to its opposite side.
The law of sines is used to solve triangles, given
1. Two angles and any side (ASA or AAS), or
2. Two sides and an angle opposite one of them (SSA).
We apply the law of sines to the easier ASA and AAS cases first, then we
turn to the more challenging SSA case.
Solving the ASA and AAS Cases
EXAMPLE
1
FIGURE 3
Solving the ASA Case
Solve the triangle in Figure 3.
7-1 Law of Sines
Solution
Solve for ␥
507
We are given two angles and the included side, which is the ASA case. Find the
third angle, then solve for the other two sides using the law of sines.
␣ ⫹ ␤ ⫹ ␥ ⫽ 180°
␥ ⫽ 180° ⫺ (␣ ⫹ ␤)
⫽ 180° ⫺ (28°0⬘ ⫹ 45°20⬘)
⫽ 106°40⬘
Solve for a
sin ␣ sin ␥
⫽
a
c
a⫽
⫽
c sin ␣
sin ␥
120 sin 28°0⬘
sin 106°40⬘
⫽ 58.8 meters
Solve for b
sin ␤ sin ␥
⫽
b
c
b⫽
⫽
c sin ␤
sin ␥
120 sin 45°20⬘
sin 106°40⬘
⫽ 89.1 meters
MATCHED PROBLEM
Solve the triangle in Figure 4.
1
FIGURE 4
Note that the AAS case can always be converted to the ASA case by first solving for the third angle. For the ASA or AAS case to determine a unique triangle,
the sum of the two angles must be between 0° and 180°, since the sum of all
three angles in a triangle is 180° and no angle can be zero or negative.
—Including the Ambiguous Case
Solving the SSA Case—
We now look at the case where we are given two sides and an angle opposite one
of the sides—the SSA case. This case has several possible outcomes, depending
on the measures of the two sides and the angle. Table 2 illustrates the various
possibilities.
508
7 ADDITIONAL TOPICS IN TRIGONOMETRY
T A B L E
2 SSA Variations
Number of
Triangles
␣
a
(h ⴝ b sin ␣)
Acute
0⬍a⬍h
0
(a)
Acute
a⫽h
1
(b)
Acute
h⬍a⬍b
2
(c)
Acute
aⱖb
1
(d)
Obtuse
0⬍aⱕb
0
(e)
Obtuse
a⬎b
1
(f)
Figure
Case
Table 2 need not be committed to memory. Usually a rough sketch of a particular situation will indicate which of the variations applies. The case where
h ⬍ a ⬍ b is referred to as the ambiguous case, because two triangles, one acute
and the other obtuse, are always possible.
1
EXAMPLE
2
Discuss which cases in Table 2 apply and why if in the solution process
of solving an SSA triangle with ␣ acute it is found that
(A) sin ␤ ⬎ 1
(B) sin ␤ ⫽ 1
(C) 0 ⬍ sin ␤ ⬍ 1
Solving the SSA Case
Solve the triangle(s) with ␣ ⫽ 123°, b ⫽ 23 centimeters, and a ⫽ 47
centimeters.
7-1 Law of Sines
Solution
509
From a rough sketch (Fig. 5), we see that there is only one triangle.
FIGURE 5
Solve for ␤
sin ␤ sin ␣
⫽
b
a
sin ␤ ⫽
b sin ␣ 23 sin 123°
⫽
a
47
␤ ⫽ sin⫺1
Solve for ␥
冢23 sin47123° 冣 ⫽ 24°
␣ ⫹ ␤ ⫹ ␥ ⫽ 180°
␥ ⫽ 180° ⫺ 123° ⫺ 24° ⫽ 33°
Solve for c
sin ␣ sin ␥
⫽
a
c
c⫽
MATCHED PROBLEM
a sin ␥ 47 sin 33°
⫽
⫽ 31 centimeters
sin ␣
sin 123°
Solve the triangle(s) with ␤ ⫽ 98°, a ⫽ 62 meters, and b ⫽ 88 meters.
2
EXAMPLE
Solving the SSA (Ambiguous) Case
3
Solve the triangle(s) with ␣ ⫽ 26°, a ⫽ 1.0 meter, and b ⫽ 1.8 meters.
Solution
If we try to draw a triangle with the indicated sides and angle, we find that
two triangles, I and II, are possible (Fig. 6). This is verified by the fact that
h ⬍ a ⬍ b, where h ⫽ b sin ␣ ⫽ 0.79 meters, a ⫽ 1.0 meter, and b ⫽ 1.8 meters.
FIGURE 6
510
7 ADDITIONAL TOPICS IN TRIGONOMETRY
Solve for ␤ and ␤ⴕ
We start by finding ␤ and ␤⬘ using the law of sines:
sin ␤ sin ␣
⫽
b
a
sin ␤ ⫽
b sin ␣ 1.8 sin 26°
⫽
⫽ 0.7891
a
1.0
Angle ␤ can be either obtuse or acute:
␤ ⫽ 180° ⫺ sin⫺1 0.7891
or
␤⬘ ⫽ sin⫺1 0.7891
⫽ 180° ⫺ 52° ⫽ 128°
Solve for ␥ and ␥ⴕ
⫽ 52°
We next find ␥ and ␥⬘:
␥ ⫽ 180° ⫺ (26° ⫹ 128°) ⫽ 26°
␥⬘ ⫽ 180° ⫺ (26° ⫹ 52°) ⫽ 102°
Solve for c and c⬘
Finally, we solve for c and c⬘:
sin ␣ sin ␥
⫽
a
c
c⫽
⫽
sin ␣ sin ␥⬘
⫽
a
c⬘
a sin ␥
sin ␣
1.0 sin 26°
sin 26°
⫽ 1.0 meter
c⬘ ⫽
⫽
a sin ␥⬘
sin ␣
1.0 sin 102°
sin 26°
⫽ 2.2 meters
In summary,
MATCHED PROBLEM
Triangle I:
⬘␤ ⫽ 128°
⬘␥ ⫽ 26°
⬘c ⫽ 1.0 meter
Triangle II:
␤⬘ ⫽ 52°
␥⬘ ⫽ 102°
c⬘ ⫽ 2.2 meters
Solve the triangle(s) with a ⫽ 8 kilometers, b ⫽ 10 kilometers, and ␣ ⫽ 35°.
3
The law of sines is useful in many applications, as can be seen in Example 4
and the applications in Exercise 7-1.
EXAMPLE
4
Surveying
To measure the length d of a lake (see Fig. 7), a base line AB is established
and measured to be 125 meters. Angles A and B are measured to be 41.6° and
124.3°, respectively. How long is the lake?
7-1 Law of Sines
511
FIGURE 7
Solution
Find angle C and use the law of sines.
Angle C ⫽ 180° ⫺ (124.3° ⫹ 41.6°)
⫽ 14.1°
sin 14.1° sin 41.6°
⫽
125
d
41.6°
冢sin
sin 14.1° 冣
d ⫽ 125
⫽ 341 meters
MATCHED PROBLEM
In Example 4, find the distance AC.
4
Answers to Matched Problems
1. ␥ ⫽ 101°40⬘, b ⫽ 141, c ⫽ 152
2. ␣ ⫽ 44°, ␥ ⫽ 38°, c ⫽ 55 m
3. ␤ ⫽ 134°, ␤⬘ ⫽ 46°, ␥ ⫽ 11°, ␥⬘ ⫽ 99°, c ⫽ 2.7 km, c⬘ ⫽ 14 km
4. 424 m
EXERCISE 7-1
The labeling in the figure below is the convention we will
follow in this exercise set. Your answers to some problems may
differ slightly from those in the book, depending on the order
in which you solve for the sides and angles of a given triangle.
4. ␤ ⫽ 43°, ␥ ⫽ 36°, a ⫽ 92 millimeters
5. ␤ ⫽ 112°, ␥ ⫽ 19°, c ⫽ 23 yards
6. ␣ ⫽ 52°, ␥ ⫽ 105°, c ⫽ 47 meters
7. ␣ ⫽ 52°, ␥ ⫽ 47°, a ⫽ 13 centimeters
8. ␤ ⫽ 83°, ␥ ⫽ 77°, c ⫽ 25 miles
In Problems 9–16, determine whether the information in each
problem allows you to construct 0, 1, or 2 triangles. Do not
solve the triangle. Explain which case in Table 2 applies.
A
9. a ⫽ 2 inches, b ⫽ 4 inches, ␣ ⫽ 30°
10. a ⫽ 3 feet, b ⫽ 6 feet, ␣ ⫽ 30°
Solve each triangle in Problems 1–8.
11. a ⫽ 6 inches, b ⫽ 4 inches, ␣ ⫽ 30°
1. ␣ ⫽ 73°, ␤ ⫽ 28°, c ⫽ 42 feet
12. a ⫽ 8 feet, b ⫽ 6 feet, ␣ ⫽ 30°
2. ␣ ⫽ 41°, ␤ ⫽ 33°, c ⫽ 21 centimeters
13. a ⫽ 1 inch, b ⫽ 4 inches, ␣ ⫽ 30°
3. ␣ ⫽ 122°, ␥ ⫽ 18°, b ⫽ 12 kilometers
14. a ⫽ 2 feet, b ⫽ 6 feet, ␣ ⫽ 30°
512
7 ADDITIONAL TOPICS IN TRIGONOMETRY
15. a ⫽ 3 inches, b ⫽ 4 inches, ␣ ⫽ 30°
34. (A) Use the law of sines and suitable identities to show
that for any triangle
16. a ⫽ 5 feet, b ⫽ 6 feet, ␣ ⫽ 30°
a⫺b
⫽
a⫹b
B
Solve each triangle in Problems 17–30. If a problem has no
solution, say so.
␣⫺␤
2
␣⫹␤
tan
2
tan
(B) Verify the formula with values from Problem 1.
17. ␣ ⫽ 118.3°, ␥ ⫽ 12.2°, b ⫽ 17.3 feet
APPLICATIONS
18. ␤ ⫽ 27.5°, ␥ ⫽ 54.5°, a ⫽ 9.27 inches
19. ␣ ⫽ 67.7°, ␤ ⫽ 54.2°, b ⫽ 123 meters
35. Coast Guard. Two lookout posts, A and B (10.0 miles
apart), are established along a coast to watch for illegal
ships coming within the 3-mile limit. If post A reports a
ship S at angle BAS ⫽ 37°30⬘ and post B reports the same
ship at angle ABS ⫽ 20°0⬘, how far is the ship from post
A? How far is the ship from the shore (assuming the shore
is along the line joining the two observation posts)?
20. ␣ ⫽ 122.7°, ␤ ⫽ 34.4°, b ⫽ 18.3 kilometers
21. ␣ ⫽ 46.5°, a ⫽ 7.9 millimeters, b ⫽ 13.1 millimeters
22. ␣ ⫽ 26.3°, a ⫽ 14.7 inches, b ⫽ 35.2 inches
23. ␤ ⫽ 38.9°, a ⫽ 42.7 inches, b ⫽ 30.0 inches, ␣ acute
24. ␤ ⫽ 27.3°, a ⫽ 244 centimeters, b ⫽ 135 centimeters,
␣ acute
36. Fire Lookout. A fire at F is spotted from two fire lookout
stations, A and B, which are 10.0 miles apart. If station B
reports the fire at angle ABF ⫽ 53°0⬘ and station A reports
the fire at angle BAF ⫽ 28°30⬘, how far is the fire from
station A? From station B?
25. ␤ ⫽ 38.9°, a ⫽ 42.7 inches, b ⫽ 30.0 inches, ␣ obtuse
26. ␤ ⫽ 27.3°, a ⫽ 244 centimeters, b ⫽ 135 centimeters,
␣ obtuse
★
37. Natural Science. The tallest trees in the world grow in
Redwood National Park in California; they are taller than
a football field is long. Find the height of one of these
trees, given the information in the figure. (The 100-foot
measurement is accurate to three significant digits.)
★
38. Surveying. To measure the height of Mt. Whitney in California, surveyors used a scheme like the one shown in the
figure in Problem 37. They set up a horizontal baseline
2,000 feet long at the foot of the mountain and found the
angle nearest the mountain to be 43°5⬘; the angle farthest
from the mountain was found to be 38°0⬘. If the baseline
was 5,000 feet above sea level, how high is Mt. Whitney
above sea level?
27. ␣ ⫽ 123.2°, a ⫽ 101 yards, b ⫽ 152 yards
28. ␣ ⫽ 137.3°, a ⫽ 13.9 meters, b ⫽ 19.1 meters
29. ␤ ⫽ 29°30⬘, a ⫽ 43.2 millimeters, b ⫽ 56.5 millimeters
30. ␤ ⫽ 33°50⬘, a ⫽ 673 meters, b ⫽ 1,240 meters
C
31. Let ␣ ⫽ 42.3° and b ⫽ 25.2 centimeters. Determine a value
k so that if 0 ⬍ a ⬍ k, there is no solution; if a ⫽ k, there is
one solution; and if k ⬍ a ⬍ b, there are two solutions.
32. Let ␣ ⫽ 37.3° and b ⫽ 42.8 centimeters. Determine a value
k so that if 0 ⬍ a ⬍ k, there is no solution; if a ⫽ k, there is
one solution; and if k ⬍ a ⬍ b, there are two solutions.
33. Mollweide’s equation,
(a ⫺ b) cos
␥
␣⫺␤
⫽ c sin
2
2
is often used to check the final solution of a triangle, since
all six parts of a triangle are involved in the equation. If
the left side does not equal the right side after substitution,
then an error has been made in solving a triangle. Use this
equation to check Problem 1. (Because of rounding errors,
both sides may not be exactly the same.)
7-1 Law of Sines
513
39. Engineering. A 4.5-inch piston rod joins a piston to a 1.5inch crankshaft (see the figure). How far is the base of the
piston from the center of the crankshaft (distance d) when
the rod makes an angle of 9° with the centerline? There
are two answers to the problem.
40. Engineering. Repeat Problem 39 if the piston rod is 6.3
inches, the crankshaft is 1.7 inches, and the angle is 11°.
41. Astronomy. The orbits of the Earth and Venus are approximately circular, with the sun at the center. A sighting of
Venus is made from Earth, and the angle ␣ is found to be
18°40⬘. If the radius of the orbit of the Earth is 1.495 ⫻
108 kilometers and the radius of the orbit of Venus is 1.085
⫻ 108 kilometers, what are the possible distances from the
Earth to Venus? (See the figure.)
42. Astronomy. In Problem 41, find the maximum angle ␣.
[Hint: The angle is maximum when a straight line joining
the Earth and Venus is tangent to Venus’s orbit.]
★
★
43. Surveying. A tree growing on a hillside casts a 102-foot
shadow straight down the hill (see the figure). Find the
vertical height of the tree if, relative to the horizontal, the
hill slopes 15.0° and the angle of elevation of the sun is
62.0°.
44. Surveying. Find the height of the tree in Problem 43 if the
shadow length is 157 feet and, relative to the horizontal,
the hill slopes 11.0° and the angle of elevation of the sun
is 42.0°.
★
45. Life Science. A cross section of the cornea of an eye, a
circular arc, is shown in the figure. Find the arc radius R
and the arc length s, given the chord length C ⫽ 11.8 millimeters and the central angle ␪ ⫽ 98.9°.
★
46. Life Science. Referring to the figure, find the arc radius R
and the arc length s, given the chord length C ⫽ 10.2 millimeters and the central angle ␪ ⫽ 63.2°.
★
47. Surveying. The procedure illustrated in Problems 37 and
38 is used to determine an inaccessible height h when a
baseline d on a line perpendicular to h can be established
(see the figure) and the angles ␣ and ␤ can be measured.
Show that
h⫽d
冤 sinsin(␤␣ sin⫺ ␣)␤ 冥
514
★★
7 ADDITIONAL TOPICS IN TRIGONOMETRY
48. Surveying. The layout in the figure is used to determine
an inaccessible height h when a baseline d in a plane perpendicular to h can be established and the angles ␣, ␤, and
␥ can be measured. Show that
h ⫽ d sin ␣ csc (␣ ⫹ ␤) tan ␥
Section 7-2 Law of Cosines
Law of Cosines Derivation
Solving the SAS Case
Solving the SSS Case
If in a triangle two sides and the included angle are given (SAS), or three sides
are given (SSS), the law of sines cannot be used to solve the triangle—neither
case involves an angle and its opposite side (Fig. 1). Both cases can be solved
starting with the law of cosines, which is the subject matter for this section.
FIGURE 1
(a) SAS case
(b) SSS case
Law of Cosines Derivation
Theorem 1 states the law of cosines.
LAW OF COSINES
1
a2 ⫽ b2 ⫹ c2 ⫺ 2bc cos ␣
b ⫽ a ⫹ c ⫺ 2ac cos ␤
2
2
2
c2 ⫽ a2 ⫹ b2 ⫺ 2ab cos ␥
All three
equations
say
essentially
the same
thing.
Cases SAS and SSS are most readily solved by starting with the law of
cosines.