Download SECTION 5-4 Trigonometric Functions

366
★
5 Trigonometric Functions
78. Engineering. In Problem 77, through what angle in radians
will the back wheel turn if the front wheel turns through 15
radians?
The arc length on a circle is easy to compute if the corresponding central angle is given in radians and the radius of
the circle is known (s r). If the radius of a circle is
large and a central angle is small, then an arc length is
often used to approximate the length of the corresponding
chord, as shown in the figure. If an angle is given in degree
measure, converting to radian measure first may be helpful
in certain problems. This information will be useful in Problems 79–82.
c
s
82. Photography. The angle of view of a 300-mm lens is 8°.
At 500 ft, what is the width of the field of view to the
nearest foot?
r
5-4
80. Astronomy. The moon is about 381,000 kilometers from
the Earth. If the angle subtended by the diameter of the
moon on the surface of the Earth is 0.0092 rad, approximately what is the diameter of the moon to the nearest hundred kilometers?
81. Photography. The angle of view of a 1000-mm telephoto
lens is 2.5°. At 750 ft, what is the width of the field of view
to the nearest foot?
c s r
SECTION
79. Astronomy. The sun is about 9.3 107 mi from the Earth.
If the angle subtended by the diameter of the sun on the surface of the Earth is 9.3 103 rad, approximately what is
the diameter of the sun to the nearest thousand miles in
standard decimal notation?
Trigonometric Functions
•
•
•
•
•
Definition of the Trigonometric Functions
Calculator Evaluation of Trigonometric Functions
Definition of the Trigonometric Functions—Alternate Form
Exact Values for Special Angles and Real Numbers
Summary of Special Angle Values
In this section we define trigonometric functions with angle domains, where angles
can have either degree or radian measure. We also show how circular functions are
related to trigonometric functions so that you will be able to move easily from one
to the other, as needed.
• Definition of the
Trigonometric
Functions
We are now ready to define trigonometric functions with angle domains. Since we
have already defined the circular functions with real number domains, we can take
advantage of these results and define the trigonometric functions with angle domains
in terms of the circular functions. To each of the six circular functions we associate
a trigonometric function of the same name. If is an angle, in either radian or degree
measure, we assign values to sin , cos , tan , csc , sec , and cot as given in
Definition 1.
5-4
DEFINITION 1
Trigonometric Functions
367
Trigonometric Functions with Angle Domains
If is an angle with radian measure x, then the value of each trigonometric
function at is given by its value at the real number x.
Trigonometric
Function
sin cos tan csc sec cot Circular
Function
sin x
cos x
tan x
csc x
sec x
cot x
b
(a, b)
W(x)
x units
arc length
x rad
(1, 0)
a
If is an angle in degree measure, convert to radian measure and proceed as
above.
[Note: To reduce the number of different symbols in certain figures, the u and
v axes we started with will often be labeled as the a and b axes, respectively.
Also, an expression such as sin 30° denotes the sine of the angle whose measure is 30°.]
The figure in Definition 1 makes use of the important fact that in a unit circle
the arc length s opposite an angle of x radians is x units long, and vice versa:
s r 1 x x
EXAMPLE 1
Exact Evaluation for Special Angles
Evaluate exactly without a calculator:
3
(A) sin
radians
(B) tan
radians
6
4
Solution
(C) cos 180°
6 radians sin 6 21
3
3
(B) tan radians tan
1
4
4
(A) sin
(C) cos 180°
(D) csc (150°)
cos ( radians)
csc cos 1
5
radians
6
56 2
csc (D) csc (150°)
368
5 Trigonometric Functions
Matched Problem 1
Evaluate exactly without a calculator:
(A) tan (/4 radians)
(B) cos (2/3 radians)
(C) sin 90°
(D) sec (120°)
• Calculator
Evaluation of
Trigonometric
Functions
How do we evaluate trigonometric functions for arbitrary angles? Just as a calculator
can be used to approximate circular functions for arbitrary real numbers, a calculator
can be used to approximate trigonometric functions for arbitrary angles.
Most calculators have a choice of three trigonometric modes: degree (decimal),
radian, or grad.
The measure of a right angle 90° radians 100 grads
2
The grad unit is used in certain engineering applications and will not be used in this
book. We repeat a caution stated earlier:
CAUTION
Read the instruction book accompanying your calculator to determine how to
put your calculator in degree or radian mode. Forgetting to set the correct
mode before starting calculations involving trigonometric functions is a frequent cause of error when using a calculator.
Using a calculator with degree and radian modes, we can evaluate trigonometric
functions directly for angles in either degree or radian measure without having to convert degree measure to radian measure first. (Some calculators work only with decimal degrees, and others work with either decimal degrees or degree–minute–second
forms. Consult your manual.)
We generalize the reciprocal identities (stated first in Theorem 1, Section 5-2) to
evaluate cosecant, secant, and cotangent.
Theorem 1
Reciprocal Identities
For x any real number or angle in degree or radian measure:
csc x 1
sin x
sin x 0
sec x 1
cos x
cos x 0
cot x 1
tan x
tan x 0
5-4
EXAMPLE 2
Trigonometric Functions
369
Calculator Evaluation
Evaluate to 4 significant digits using a calculator:
(A) cos 173.42°
(B) sin (3 radians)
(C) tan 7.183
(D) cot (102°51)
(E) sec (12.59 radians)
(F) csc (206.3)
Solutions
(A) cos 173.42° 0.9934
(B) sin (3 radians) 0.1411
(C) tan 7.183 1.260
Degree mode
Radian mode
Radian mode
cot (102.85°)
(D) cot (102°51)
0.2281
(E) sec (12.59 radians) 1.000
(F) csc (206.3) 1.156
Matched Problem 2
Radian mode
Radian mode
Evaluate to 4 significant digits using a calculator:
(A) sin 239.12°
(D) tan (212°33)
• Definition of the
Trigonometric
—Alternate
Functions—
Form
Degree mode (Some calculators require
decimal degrees.)
(B) cos (7 radians)
(E) sec (8.09 radians)
(C) cot 10
(F) csc (344.5)
For many applications involving the use of trigonometric functions, including triangle applications, it is useful to write Definition 1 in an alternate form—a form that
utilizes the coordinates of an arbitrary point (a, b) (0, 0) on the terminal side of
an angle (see Fig. 1).
This alternate form of Definition 1 is easily found by inserting a unit circle in
Figure 1, drawing perpendiculars from points P and Q to the horizontal axis (Fig. 2),
and utilizing the fact that ratios of corresponding sides of similar triangles are proportional.
b
Q(a, b)
P(a, b)
x units
b
x rad
O
P(a, b)
(1, 0)
O
FIGURE 1 Angle .
a
FIGURE 2 Similar triangles.
a
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5 Trigonometric Functions
Letting r d(O, P) and noting that d(O, Q) 1, we have
sin sin x b b b
1
r
b and b always have the same sign.
cos cos x a a a
1
r
a and a always have the same sign.
The values of the other four trigonometric functions can be obtained using basic identities. For example,
tan sin b/r b
cos a/r a
We now have the very useful alternate form of Definition 1 given below.
DEFINITION 1
(ALTERNATE FORM)
Trigonometric Functions with Angle Domains
If is an arbitrary angle in standard position in a rectangular coordinate system and P(a, b) is a point r units from the origin on the terminal side of , then:
b
b
a
P (a, b)
r
a
b
b
a
a
r
P (a, b)
sin b
r
r
csc ,
b
b
0
cos a
r
r
sec ,
a
a
0
a
cot ,
b
b
0
b
tan ,
a
b
a
0
a
a
b
r
P (a, b)
r a2 b2 0;
Domains: Sets of all possible angles for which the ratios are defined
Ranges: Subsets of the set of real numbers
(Domains and ranges will be stated more precisely in Section 5-6.)
[Note: The right triangle formed by drawing a perpendicular from P(a, b) to the
horizontal axis is called the reference triangle associated with the angle . We
will often refer to this triangle.]
5-4
EXPLORE-DISCUSS 1
Trigonometric Functions
371
Discuss why, for a given angle , the ratios in Definition 1 are independent of the
choice of P(a, b) on the terminal side of as long as (a, b) (0, 0).
The alternate form of Definition 1 should be memorized. As a memory aid, note
that when r 1, then P(a, b) is on the unit circle, and all function values correspond
to the values obtained using Definition 1 for circular functions in Section 5-2. In fact,
using the alternate form of Definition 1 in conjunction with the original statement of
Definition 1 in this section, we have an alternate way of evaluating circular functions:
Circular Functions and Trigonometric Functions
For x any real number:
sin x sin (x radians)
cos x cos (x radians)
sec x sec (x radians)
csc x csc (x radians)
tan x tan (x radians)
cot x cot (x radians)
(1)
Thus, we are now free to evaluate circular functions in terms of trigonometric
functions, using reference triangles where appropriate, or in terms of circular points
and the wrapping function discussed earlier. Each approach has certain advantages in
particular situations, and you should become familiar with the uses of both approaches.
It is because of equations (1) that we are able to evaluate circular functions using
a calculator set in radian mode (see Section 5-2). Generally, unless a certain emphasis is desired, we will not use “rad” after a real number. That is, we will interpret
expressions such as “sin 5.73” as the “circular function value sin 5.73” or the “trigonometric function value sin (5.73 rad)” by the context in which the expression occurs
or the form we wish to emphasize. We will remain flexible and often switch back and
forth between circular function emphasis and trigonometric function emphasis,
depending on which approach provides the most enlightenment for a given situation.
EXAMPLE 3
Evaluating Trigonometric Functions
Find the value of each of the six trigonometric functions for the illustrated angle with terminal side that contains P(3, 4). See Figure 3.
FIGURE 3
b
5
a
5
P (3, 4)
5
r
5
372
5 Trigonometric Functions
(a, b) (3, 4)
Solution
r a b (3)2 (4)2 25 5
2
Matched Problem 3
EXAMPLE 4
2
sin b 4
4
r
5
5
csc r
5
5
b 4
4
cos a 3
3
r
5
5
sec r
5
5
a 3
3
tan b 4 4
a 3 3
cot a 3 3
b 4 4
Find the value of each of the six trigonometric functions if the terminal side of contains the point (6, 8). [Note: This point lies on the terminal side of the angle
in Example 3; hence, the final results should be the same as those obtained in
Example 3.]
Evaluating Trigonometric Functions
Find the value of each of the other five trigonometric functions for an angle (without finding ) given that is a IV quadrant angle and sin 45.
Solution
The information given is sufficient for us to locate a reference triangle in quadrant
IV for , even though we don’t know what is. We sketch a reference triangle, label
what we know (Fig. 4), and then complete the problem as indicated.
FIGURE 4
b
5
a
5
5
5
Since sin b/r 45 , we can let b 4 and r 5
(r is never negative). If we can find a, then we can
determine the values of the other five functions.
a
4
P (a, 4)
5
Terminal side of Use the Pythagorean theorem to find a:
a2 (4)2 52
a2 9
a 3
3
a cannot be negative because is a IV quadrant angle.
Using (a, b) (3, 4) and r 5, we have
5-4
cos a 3
r 5
tan b 4
4
a
3
3
sec r 5
a 3
cot 373
Trigonometric Functions
csc r
5
5
b 4
4
a
3
3
b 4
4
Matched Problem 4
Find the value of each of the other five trigonometric functions for an angle (without finding ) given that is a II quadrant angle and tan 43 .
• Exact Values for
Special Angles and
Real Numbers
Assuming a trigonometric function is defined, it can be evaluated exactly without the
use of a calculator or table (which is different from finding approximate values using
a calculator or table) for any integer multiple of 30°, 45°, 60°, 90°, /6, /4, /3, or
/2. With a little practice you will be able to determine these values mentally. Working with exact values has advantages over working with approximate values in many
situations.
The easiest angles to deal with are quadrantal angles since these angles are integer multiples of 90° or /2. It is easy to find the coordinates of a point on a coordinate axis. Since any nonorigin point will do, we shall for convenience choose points
1 unit from the origin, as shown in Figure 5.
FIGURE 5 Quadrantal angles.
b
(0, 1)
(1, 0)
(1, 0)
a
In each case, r a2 b2 1, a positive number.
(0, 1)
EXAMPLE 5
Trig Functions of Quadrantal Angles
Find:
(A) sin 90°
Solutions
(B) cos (C) tan (2)
(D) cot (180°)
For each, visualize the location of the terminal side of the angle relative to Figure 3.
With a little practice, you should be able to do most of the following mentally.
(A) sin 90°
(B) cos b
r
a
r
1
1
1
1
1
1
b
(a, b) (0, 1); r 1
a
b
(a, b) (1, 0); r 1
a
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5 Trigonometric Functions
b
b
a
(C) tan (2)
0
0
1
1
0
a
b
(D) cot (180°)
(a, b) (1, 0); r 1
a
b
(a, b) (1, 0); r 1
a
Not defined
Matched Problem 5
Find:
(A) sin (3/2)
EXPLORE-DISCUSS 2
(B) sec ()
(C) tan 90°
(D) cot (270°)
Notice in Example 5D that cot (180°) is not defined. Discuss other angles in
degree measure for which the cotangent is not defined. For what angles in degree
measure is the cosecant function not defined?
Because the concept of reference triangle introduced in Definition 1 (alternate
form) plays an important role in much of the material that follows, we restate its definition here and define the related concept of reference angle.
Reference Triangle and Reference Angle
1. To form a reference triangle for , draw a perpendicular from a point
P(a, b) on the terminal side of to the horizontal axis.
2. The reference angle is the acute angle (always taken positive) between
the terminal side of and the horizontal axis.
b
a
a
(a, b) (0, 0)
is always positive
b
P (a, b)
Figure 6 shows several reference triangles and reference angles corresponding to
particular angles.
5-4
Trigonometric Functions
375
Reference
triangle
FIGURE 6 Reference triangles and
reference angles.
Reference
angle
Reference
angle
45
120
180
Reference
triangle
90
(a)
180° 120° 60°
(b)
45°
/2
5/4
/6 (c)
5
4
4
(d)
180
360
6
7/6
420
(e)
420° 360° 60°
(f)
7
6
6
If a reference triangle of a given angle is a 30°–60° right triangle or a 45° right
triangle, then we can find exact coordinates, other than (0, 0), on the terminal side of
the given angle. To this end, we first note that a 30°–60° triangle forms half of an
equilateral triangle, as indicated in Figure 7. Because all sides are equal in an equilateral triangle, we can apply the Pythagorean theorem to obtain a useful relationship
among the three sides of the original triangle:
c 2a
FIGURE 7 30°–60° right triangle.
b c2 a2
30 30
c
c
60
a
a
c
30
(/6)
2a
3a2 a3
b
60
(2a)2 a2
60
(/3)
a
a3
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5 Trigonometric Functions
Similarly, using the Pythagorean theorem on a 45° right triangle, we obtain the
result shown in Figure 8.
c a2 a2
FIGURE 8 45° right triangle.
2a2
45
c
a2
a2
a
45
(/4)
a
45
(/4)
a
45
a
Figure 9 illustrates the results shown in Figures 7 and 8 for the case a 1. This
case is the easiest to remember. All other cases can be obtained from this special case
by multiplying or dividing the length of each side of a triangle in Figure 9 by the
same nonzero quantity. For example, if we wanted the hypotenuse of a special 45°
right triangle to be 1, we would simply divide each side of the 45° triangle in Figure
9 by 2.
– 60° and 45° Special Triangles
30°–
FIGURE 9
30
(/6)
2
3
45
(/4)
2
60
(/3)
1
1
45
(/4)
1
If an angle or a real number has a 30°–60° or a 45° reference triangle, then we
can use Figure 9 to find exact coordinates of a nonorigin point on the terminal side
of the angle. Using the definition of the trigonometric functions, Definition 1 alternate form, we will then be able to find the exact value of any of the six functions for
the indicated angle or real number.
EXAMPLE 6
Exact Evaluation
Evaluate exactly using appropriate reference triangles:
(A) cos 60°, sin (/3), tan (/3)
Solutions
(B) sin 45°, cot (/4), sec (/4)
(A) Use the special 30°–60° triangle with sides 1, 2, and 3 as the reference triangle, and use 60° or /3 as the reference angle (Fig. 10). Use the sides of the reference triangle to determine P(a, b) and r; then use the appropriate definitions.
5-4
FIGURE 10
b
cos 60 (a, b) (1, 3)
r2
2
60
(/3)
b 3
3
r
2
tan
b 3
3
3
a
1
a
1
a 1
r 2
sin
3
377
Trigonometric Functions
(B) Use the special 45° triangle with sides 1, 1, and 2 as the reference triangle,
and use 45° or /4 as the reference angle (Fig. 11). Use the sides of the reference triangle to determine P(a, b) and r; then use the appropriate definitions.
FIGURE 11
b
sin 45 (a, b) (1, 1)
r 2
2
a 1
1
4
b 1
sec
r 2
2
4
a
1
a
1
Matched Problem 6
cot
1
45
(/4)
b
1
2
or
r 2
2
Evaluate exactly using appropriate reference triangles:
(A) cos 45°, tan (/4), csc (/4)
(B) sin 30°, cos (/6), cot (/6)
Before proceeding, it is useful to observe from a geometric point of view multiples of /3 (60°), /6 (30°), and /4 (45°). These are illustrated in Figure 12.
3
6
2
3
3
4
6
2
2
3
2
4
2
6
6
3
0
2
6
6
12
6
7
6
4
3
5
3
8
6
11
6
4
3
10
6
9
6
Multiples of /3 (60)
( ) of special angles.
FIGURE 12 Multiples
3
2
Multiples of /6 (30)
5
3
2
4
3
4
6
5
6
3
3
3
0
2
4
4
8
4
5
4
7
4
6
4
3
2
Multiples of /4 (45)
0
2
378
5 Trigonometric Functions
EXAMPLE 7
Exact Evaluation
Evaluate exactly using appropriate reference triangles:
(A) cos (7/4)
Solutions
(B) sin (2/3)
(C) tan 210°
(D) sec (240°)
Each angle (or real number) has a 30°–60° or 45° reference triangle. Locate it, determine (a, b) and r as in Example 6, and then evaluate.
(A) cos
7
1
2
or
4
2
2
(B) sin
2 3
3
2
b
b
7
4
1
(a, b) (1, 3)
r2
a
4
2
3
1
2
(a, b) (1, 1)
r 2
(C) tan 210° 1
1
3
or
3 3
3
(D) sec (240°) 1
b
a
2
3
(a, b) (3, 1)
r2
Matched Problem 7
2
2
1
(a, b) (1, 3)
r2
210
30
a
1
b
3
2
3
3
2
60
1
a
240
Evaluate exactly using appropriate triangles:
(A) tan (/4)
(B) sin 210°
(C) cos (2/3)
(D) csc (240°)
Now the problem is reversed; that is, let the exact value of one of the six trigonometric functions be given and assume this value corresponds to one of the special reference triangles. Can a smallest positive be found for which the trigonometric function has that value? Example 8 shows how.
5-4
EXAMPLE 8
Trigonometric Functions
379
Finding Special Angles
Find the smallest positive in degree and radian measure for which each is true.
(A) tan 1/3
Solutions
(A)
(B) sec 2
b
1
a 3
We can let (a, b) (3, 1) or (3, 1). The smallest positive for which
this is true is a quadrant I angle with reference triangle as drawn in Figure 13.
tan b
FIGURE 13
30° or
6
(3, 1)
1
30
a
3
sec (B)
r 2
a 1
Because r 0
a is negative in quadrants II and III. The smallest positive is associated with
a 45° reference triangle in quadrant II as drawn in Figure 14.
b
FIGURE 14
135° or
3
4
2
1
45
1
Matched Problem 8
a
Find the smallest positive in degree and radian measure for which each is true.
(A) sin 3/2
(B) cos 1/2
Remark. After quite a bit of practice, the reference triangle figures in Examples 7
and 8 can be visualized mentally; however, when in doubt, draw a figure.
380
5 Trigonometric Functions
• Summary of
Special Angle Values
TABLE 1
Table 1 includes a summary of the exact values of the sine, cosine, and tangent for
the special angle values from 0° to 90°. Some people like to memorize these values,
while others prefer to memorize the triangles in Figure 9. Do whichever is easier for
you.
Special Angle Values
sin cos tan 0°
0
1
0
30°
1
2
3/2
1/ 3 or 3/3
45°
1/ 2 or 2/2
1/ 2 or 2/2
1
60°
3/2
1
2
3
90°
1
0
Not defined
These special angle values are easily remembered for sine and cosine if you note
the unexpected pattern after completing Table 2 in Explore-Discuss 3.
EXPLORE-DISCUSS 3
TABLE 2
Fill in the cosine column in Table 2 with a pattern of values that is similar to those
in the sine column. Discuss how the two columns of values are related.
Special Angle Values—Memory Aid
sin cos 0°
0/2 0
1
2
30°
1/2 45°
2/2
60°
3/2
90°
4/2 1
Cosecant, secant, and cotangent can be found for these special angles by using
the values in Tables 1 or 2 and the reciprocal identities from Theorem 1.
Answers to Matched Problems
(A) 1
(B) 12
(C) 1
(D) 2
(A) 0.8582
(B) 0.7539
(C) 1.542
(D) 0.6383
(E) 4.277
sin 45 , cos 35 , tan 34 , csc 45 , sec 53 , cot 43
sin 35 , cos 45 , csc 35 , sec 45 , cot 43
(A) 1
(B) 1
(C) Not defined
(D) 0
(A) cos 45° 1/2, tan (/4) 1, csc (/4) 2
(B) sin 30° 12 , cos (/6) 3/2, cot (/6) 3
7. (A) 1
(B) 12
(C) 12
(D) 2/3
(E) 3
(F) 2
8. (A) 60° or /3
(B) 135° or 3/4
1.
2.
3.
4.
5.
6.
(F) 1.137
5-4
EXERCISE
Find the value of each of the six trigonometric functions for
an angle that has a terminal side containing the point
indicated in Problems 1–4.
1. (6, 8)
2. (3, 4)
3. (1, 3)
4. (3, 1)
5. sin 68°
6. tan 21°
7. cot 5
8. csc (11)
9. cos 78.24°
11. csc 365°5248
12. sec 88°2715
13. tan (1.58)
14. cot 25.1
In Problems 15–26, evaluate exactly, using reference triangles where appropriate, without using a calculator.
15. sin 0°
16. cos 0°
17. tan 60°
18. cos 30°
19. sin 45°
20. csc 60°
21. sec 45°
22. cot 45°
23. cot 0°
24. cot 90°
25. tan 90°
26. sec 0°
Find the reference angle for each angle in Problems
27–32.
4
31. 5
3
42. tan (135°)
43. cos (13/6)
44. sec (13/4)
45. tan (7/3)
46. cos (2/3)
47. sec (23/4)
48. csc (17/6)
29. 49. cos 50. sec 51. tan 52. cot 53. csc 54. sin In Problems 55–60, find the smallest positive in degree
and radian measure for which:
10. sin 45.01°
28. 135°
41. csc (150°)
For which values of , 0° 360°, is each of Problems
49–54 not defined? Explain why.
Evaluate Problems 5–14 to 4 significant digits using a calculator. Make sure your calculator is in the correct mode
(degree or radian) for each problem.
30. 381
5-4
A
27. 300°
Trigonometric Functions
7
6
32. 5
4
55. cos 1
2
56. sin 57. sin 1
2
58. tan 3
59. csc 2
3
60. sec 2
3
2
Find the value of each of the other five trigonometric functions for an angle , without finding , given the information
indicated in Problems 61–64. Sketching a reference triangle
should be helpful.
61. sin 35 and cos 0
62. tan 43 and sin 0
63. cos 5/3 and cot 0
64. cos 5/3 and tan 0
65. Which trigonometric functions are not defined when the
terminal side of an angle lies along the vertical axis? Why?
B
66. Which trigonometric functions are not defined when the
terminal side of an angle lies along the horizontal axis?
Why?
In Problems 33–48, evaluate exactly, using reference angles
where appropriate, without using a calculator.
67. Find exactly, all , 0° 360°, for which
cos 3/2.
33. tan (3/4)
34. cos (7/6)
35. sin (30°)
36. cot 120°
68. Find exactly, all , 0° 360°, for which
cot 1/3.
37. sec (5/6)
38. csc (5/4)
69. Find exactly, all , 0 2, for which tan 1.
39. cot 315°
40. sin 240°
70. Find exactly, all , 0 2, for which sec 2.
382
5 Trigonometric Functions
C
For Problems 71 and 72, refer to the following figure.
77. Physics—Engineering. The figure illustrates a piston connected to a wheel that turns 3 revolutions per second;
hence, the angle is being generated at 3(2) 6 radians
per second, or 6t, where t is time in seconds. If P is at
(1, 0) when t 0, show that
y b 42 a2
s
P(a, b)
sin 6t 16 (cos 6t)2
for t 0.
A
y
71. If the coordinates of A are (4, 0) and arc length s is 7 units,
find:
(A) The exact radian measure of (B) The coordinates of P to three decimal places
y
72. If the coordinates of A are (2, 0) and arc length s is 8 units,
find:
(A) The exact radian measure of (B) The coordinates of P to three decimal places
4 inches
3 revolutions
per second
a
73. In a rectangular coordinate system, a circle with its center
at the origin passes through the point (63, 6). What is the
length of the arc on the circle in quadrant I between the positive horizontal axis and the point (63, 6)?
P (a, b)
b
(1, 0)
74. In a rectangular coordinate system, a circle with its center
at the origin passes through the point (2, 23). What is the
length of the arc on the circle in quadrant I between the positive horizontal axis and the point (2, 23)?
x
6t
78. Physics—Engineering. In Problem 77, find the position of
the piston y when t 0.2 second (to 3 significant digits).
★ 79.
APPLICATIONS
75. Solar Energy. The intensity of light I on a solar cell
changes with the angle of the sun and is given by the
formula I k cos , where k is a constant (see the figure).
Geometry. The area of a regular n-sided polygon circumscribed about a circle of radius 1 is given by
A n tan
180
n
Sun
r1
Solar cell
Find light intensity I in terms of k for 0°, 30°, and
60°.
76. Solar Energy. Refer to Problem 75.
Find light intensity I in terms of k for 20°, 50°, and
90°.
n8
(A) Find A for n 8, n 100, n 1,000, and n 10,000.
Compute each to five decimal places.
(B) What number does A seem to approach as n → ?
(What is the area of a circle with radius 1?)
5-5
★ 80.
Geometry. The area of a regular n-sided polygon inscribed
in a circle of radius 1 is given by
A
n
360
sin
2
n
Solving Right Triangles
383
(B) Find the equation of a line passing through (4, 5)
with an angle of inclination 137°. Write the answer in
the form y mx b, with m and b to two decimal
places.
y
L
(A) Find A for n 8, n 100, n 1,000, and n 10,000.
Compute each to five decimal places.
(B) What number does A seem to approach as n → ?
(What is the area of a circle with radius 1?)
81. Angle of Inclination. Recall (Section 2-2) that the slope of
a nonvertical line passing through points P1(x1, y1) and
P2(x2, y2) is given by Slope m ( y2 y1)/(x2 x1). The
angle that the line L makes with the x axis, 0° 180°,
is called the angle of inclination of the line L (see figure).
Thus,
Slope m tan 0° 180°
(A) Compute the slopes to two decimal places of the lines
with angles of inclination 88.7° and 162.3°.
SECTION
c
a
FIGURE 1
5-5
b
L
x
82. Angle of Inclination. Refer to Problem 81.
(A) Compute the slopes to two decimal places of the lines
with angles of inclination 5.34° and 92.4°.
(B) Find the equation of a line passing through (6, 4)
with an angle of inclination 106°. Write the answer in
the form y mx b, with m and b to two decimal
places.
Solving Right Triangles*
In the previous sections we applied trigonometric and circular functions in the solutions of a variety of significant problems. In this section we are interested in the particular class of problems involving right triangles. Referring to Figure 1, our objective is to find all unknown parts of a right triangle, given the measure of two sides
or the measure of one acute angle and a side. This is called solving a right triangle.
Trigonometric functions play a central role in this process.
To start, we locate a right triangle in the first quadrant of a rectangular coordinate system and observe, from the definition of the trigonometric functions, six
trigonometric ratios involving the sides of the triangle. (Note that the right triangle is
the reference triangle for the angle .)
Trigonometric Ratios
c
a
0° 90°
(a, b)
sin b
c
csc c
b
b
cos a
c
sec c
a
tan b
a
cot a
b
*This section provides a significant application of trigonometric functions to real-world problems. However, it may be postponed or omitted without loss of continuity, if desired. Some may want to cover the
section just before Sections 7-1 and 7-2.