Download SECTION 6-4 Exact Values for Special Angles and Real Numbers

444
6 Trigonometric Functions
65. Angle of Inclination. Recall that (Section 3-2) 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 y
L
L
x
0° 180°
Figure for 65 and 66
(A) Compute the slopes to two decimal places of the lines
with angles of inclination 88.7° and 162.3°.
(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.
SECTION
6-4
66. Angle of Inclination. Refer to Problem 65.
(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.
Exact Values for Special Angles
and Real Numbers
•
•
•
•
Quadrantal Angles
Reference Triangles and Reference Angles
Evaluation for Angles or Real Numbers with 30°60° or 45° Reference Triangles
Summary of Special Angle Values
In the last section a calculator was used to evaluate trigonometric functions. In most
cases approximate values resulted, rounded to several decimal places. If integer multiples of 30°, 45°, /6 rad, or /4 rad are chosen, or if a real number that is an integer multiple of /6 or /4 is chosen, then for those values for which a trigonometric
function is defined, the function can be evaluated exactly without the use of a calculator. With a little practice, you will be able to carry out most of these evaluations
mentally. Working with exact values has advantages over working with approximate
values in many situations.
There are many significant applications of trigonometric functions, as we will see.
Some require angle domains (in degree or radian measure), and others require real
number domains. Our definitions of these functions enable us to shift from angle
domains to real number domains, and vice versa, with relative ease.
• Quadrantal Angles
The easiest angles to work with are quadrantal angles—that is, angles with their terminal side lying along a coordinate axis. 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 point
other than the origin will do, we choose points 1 unit from the origin, as shown in
Figure 1.
6-4
445
Exact Values for Special Angles and Real Numbers
b
FIGURE 1 Quadrantal angles.
(0, 1)
(1, 0)
(1, 0)
a
In each case, r a2 b2 1,
a positive number.
(0, 1)
EXAMPLE 1
Trig Functions of Quadrantal Angles
Find:
Solutions
(B) cos (A) sin 90°
(C) tan (2)
(D) cot (180°)
For each, visualize the location of the terminal side of the angle relative to Figure 1.
With a little practice, you should be able to do most of the following mentally.
(A) sin 90°
(B) cos 1
1
1
b
r
a
r
1
1
1
b
(a, b) (0, 1); r 1
a
b
(a, b) (1, 0); r 1
a
b
(C) tan (2)
(D) cot (180°)
b
a
a
b
0
0
1
1
0
(a, b) (1, 0); r 1
a
b
(a, b) (1, 0); r 1
a
Not defined
Matched Problem 1
EXPLORE-DISCUSS 1
Find:
(A) sin (3/2)
(B) sec ()
(C) tan 90°
(D) cot (270°)
Notice that in Example 1D, cot (180°) is not defined.
1. For what other angles is the cotangent function not defined?
2. For what real numbers is the cotangent function not defined?
446
6 Trigonometric Functions
• Reference
Triangles and
Reference Angles
DEFINITION 1
Because the reference triangle is going to play a very important role in the work that
follows, we restate its definition and also define a 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)
EXAMPLE 2
Finding Reference Angles
Find the reference angle associated with each angle .
(A) 120°
(D) /6
Solutions
(B) 45°
(E) 420°
(C) 5/4
(F) 7/6
Locate angle in standard position in a coordinate system; then find the reference
angle using Definition 1. Remember, the reference angle is always positive.
(A) 180° 120° 60°
(B) 45 45
Reference
triangle
Reference
angle
Reference
angle
120
180
Reference
triangle
90
45
6-4
(C) 5/4 /4
Exact Values for Special Angles and Real Numbers
(D) /6
/2
5/4
447
/6 (E) 420° 360° 60°
(F) 7/6 /6
180
360
7/6
420
Matched Problem 2
Find the reference angle associated with each angle .
(A) 150°
(D) /4
• Evaluation for
Angles or Real
Numbers with
30°–60° or 45°
Reference Triangles
(B) 60°
(E) 390°
(C) 7/4
(F) 5/6
A 30°–60° right triangle forms half an equilateral triangle, as indicated in Figure 2.
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
b c2 a2
30 30
c
c
60
a
FIGURE 2 30°–60° right triangle.
a
c
30
(/6)
2a
3a2
b
60
(2a)2 a2
a3
60
(/3)
a
a3
448
6 Trigonometric Functions
Similarly, using the Pythagorean theorem on a 45° right triangle, we obtain the
result shown in Figure 3.
c a2 a2
FIGURE 3 45° right triangle.
2a2
45
c
a2
a2
a
45
(/4)
a
45
(/4)
a
45
a
Figure 4 illustrates the results shown in Figures 2 and 3 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 4 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
4 by 2.
If an angle or a real number has a 30°–60° or a 45° reference triangle, then we
can use Figure 4 to find exact coordinates of a nonorigin point on the terminal side
of the angle. Using the definitions of the trigonometric functions in Section 6-3, we
will then be able to find the exact value of any of the six functions for the indicated
angle or real number.
FIGURE 4
30°– 60° and 45° Special Triangles
30
(/6)
2
3
45
(/4)
2
60
(/3)
1
EXAMPLE 3
1
45
(/4)
1
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. Use the sides of the reference
triangle to determine P(a, b); then use the appropriate definitions.
6-4
b
cos 60 (a, b) (1, 3)
r2
2
b 3
3
r
2
tan
b 3
3
3
a
1
a
1
a 1
r 2
sin
3
60
(/3)
449
Exact Values for Special Angles and Real Numbers
(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. Use the sides of the reference triangle to determine P(a, b); then use the appropriate definitions.
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 3
cot
1
45
(/4)
1
b
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 multiples of /3 (60°), /6 (30°), and
/4 (45°) from a geometric point of view. These are illustrated in Figure 5.
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)
(a)
FIGURE 5 Multiples of special
angles.
3
2
Multiples of /6 (30)
(b)
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)
(c)
0
2
450
6 Trigonometric Functions
EXAMPLE 4
Exact Evaluation
Evaluate exactly using appropriate reference triangles:
(A) cos (7/4)
(D) sec (240°)
Solutions
(B) sin (2/3)
(E) csc (5/6)
(C) tan 210°
(F) cot 315°
Each angle (or real number) has a 30°–60° or a 45° reference triangle. Locate it,
determine (a, b) and r as in Example 3, and then evaluate.
(A) cos
7
1
2
or
4
2
2
(B) sin
2 3
3
2
b
b
7
4
1
a
4
(a, b) (1, 3)
r2
2
3
1
2
(a, b) (1, 1)
r 2
(C) tan 210 1
1
3
or
3 3
3
210
3
(D) sec (240) 2
60
1
5
2
2
6
1
(F) cot 315 315
3
1
2
(a, b) (3, 1)
r2
a
240
1
1
1
b
b
6
b
2
3
(a, b) (3, 1)
r2
(E) csc
2
2
1
a
30
1
a
1
(a, b) (1, 3)
r2
b
2
3
3
1
a
5
6
a
45
2
1
(a, b) (1, 1)
r 2
6-4
Matched Problem 4
Exact Values for Special Angles and Real Numbers
451
Evaluate exactly using appropriate reference triangles:
(A) tan (/4)
(D) csc (240°)
(B) sin 210°
(E) cot (5/6)
(C) cos (2/3)
(F) sec 420°
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 5 shows how.
EXAMPLE 5
Finding Special Angles
Find the smallest positive in degree and radian measure for which each is true.
(A) tan 1/3
Solutions
(B) sec 2
tan (A)
1
b
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:
b
(3, 1)
30 or
1
30
a
3
(B)
6
b
2
1
sec 45
1
r 2
a 1
Because r 0
a
a is negative in quadrants II and III. The smallest positive is associated with a
45° reference triangle in quadrant II, as drawn above:
135° or
3
4
452
6 Trigonometric Functions
Matched Problem 5
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 4
and 5 can be visualized mentally; however, when in doubt, draw a figure.
• 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 5. 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 2.
EXPLORE-DISCUSS 2
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 in
Section 6-3.
6-4
453
Exact Values for Special Angles and Real Numbers
Answers to Matched Problems
1. (A) 1
(B) 1
(C) Not defined
(D) 0
2. (A) 30°
(B) 60°
(C) /4
(D) /4
(E) 30°
(F) /6
3. (A) 1/2 or 2/2, 1, 2
(B) 12 , 3/2, 3
4. (A) 1
(B) 21
(C) 21
(D) 2/3 or 23/3
(E) 3
(F) 2
5. (A) 60° or /3
(B) 135° or 3/4
EXERCISE
6-4
A
37. sec
Find the reference angle for each angle in Problems
1–6.
40. tan 390°
1. 300°
4. 4
2. 135°
5. 5
3
3. 7
6
6. 5
4
In Problems 7–30, evaluate exactly without using a calculator or table. Leave answers involving radicals in exact radical form.
7. sin 0°
8. cos 0°
9. cos 30°
6
10. sin 45°
11. sin
13. cos
2
14. sin 90°
16. cot 30°
17. tan
4
18. sin
19. tan 90°
20. cot
2
21. sec
22. sec 0
23. csc 60°
25. cot (60°)
26. tan 28. sin (30°)
12. cos
3
23 38. csc 39. cot 270°
41. csc (240°)
42. sec (450°)
5
6
44. cot
73 47. sec 750°
46. cos 4
3
45. tan
11
4
48. sin 690°
For which values of , 0° 360°, is each of Problems
49–54 not defined? Explain why.
49. cos 50. sec 51. tan 52. cot 53. csc 54. sin For which values of x, 0 x 2, is each of Problems
55–60 not defined? Explain why.
15. cot 60°
4
29. csc 3
43. sin
7
4
55. sin x
56. tan x
57. cot x
4
58. csc x
59. sec x
60. cos x
6
C
24. cot 0
In Problems 61–66, find the smallest positive in degree
measure for which:
27. cos (90°)
61. sin 1/2
62. cos 3/2
63. tan 3
64. csc 2
65. sec 2
66. cot 1
30. sec 4
In Problems 67–72, find the smallest positive real number
for which:
B
In Problems 31–48, evaluate exactly without using a calculator or table. Leave answers involving radicals in exact
radical form.
31. tan 135°
32. sin 210°
34. cot 225°
35. sin
2
3
33. cos 300°
3 36. cos 67. sin x 3/2
68. cos x 1
69. cot x 3
70. tan x 1
71. sec x 2/3
72. csc x 2
73. Find exactly all , 0° 360°, for which cos 3/2.
74. Find exactly all , 0° 360°, for which cot 1/3.
454
6 Trigonometric Functions
75. Find exactly all , 0 2, for which tan 1.
Find exact values of x and y in Problems 83–86.
76. Find exactly all , 0 2, for which sec 2.
83. (A)
(B)
1
1
45
x
60
x
APPLICATIONS
77. Explain why the area of an n-sided regular polygon, inscribed in a circle of radius r, is given by
A nr2 cos sin
n
n
78. Explain why the area of an n-sided regular polygon, circumscribed about a circle of radius r, is given by
A nr2 tan
n
In Problems 79–82, use the indicated values of n and r, and
the formulas from Problems 77 and 78, to find the area of
an n-sided regular polygon inscribed in a circle of radius r
and the area of an n-sided regular polygon circumscribed
about a circle of radius r. Compute each area to 4 significant digits if not an integer.
84. (A)
(B)
1
y
/6
x
85. (A)
(B)
x
y
(C)
2
x
x
45
y
60
6
30
y
3
y
86. (A)
(B)
(C)
5
x
80. n 6, r 4 inches
6
81. n 8, r 10 centimeters
/3
82. n 12, r 5 meters
6-5
1
/4
x
79. n 4, r 3 inches
SECTION
y
y
x
/4
y
8
y
/6
x
y
Circular Functions
•
•
•
•
•
•
Circular Functions
Sine and Cosine Domains and Ranges
Periodic Properties
Basic Identities
Circular Functions and Trigonometric Functions
Evaluating Circular Functions
As indicated earlier, our development of the trigonometric functions proceeds from
the concrete to the abstract, which follows the historical development of the subject.
We first defined trigonometric ratios relative to right triangles, a procedure followed
by the ancient Greeks. We then expanded the meaning of these functions by using
generalized angles in standard positions in a rectangular coordinate system.
We now turn to a more abstract contemporary approach involving real number
domains without angles or triangles. This approach is preferred for advanced mathematics, including calculus, and many areas in the sciences. Using this approach we
will be able to immediately observe some very useful trigonometric properties and
relationships that are not as apparent using the angle approach.