1. (a) sœr)œ(10)ˆ ‰451 œ8 m 1 (b) sœr)œ(10)(110°)ˆ180°1 ‰œ 110181 œ 5591 m 2. )œ œsr 1081 œ 541 radians and 541ˆ180°1 ‰œ225°
3. )œ80° 80°Ê œ) ˆ180°1 ‰œ 491 Ê œ s (6)ˆ ‰491 œ8.4 in. (since the diameterœ12 in. Ê radiusœ6 in.)
4. dœ1 meter Ê œ r 50 cm Ê œ œ) sr 3050œ0.6 rad or 0.6ˆ180°1 ‰¸34°
5. 0 6
sin 0 0
cos 1 0
tan 0 3 0 und.
cot und. und. 0 1
sec 1 und. 2
csc und. und. 2
) 1
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. sin cos
tan und. 3
cot 3 3
sec und. 2
csc )
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7. cos xœ 45, tan xœ 34 8. sin xœÈ25, cos xœ È"5
9. sin xœ È83 , tan xœ È8 10. sin xœ1312, tan xœ 125 11. sin xœ È" , cos xœ È 12. cos xœ È# , tan xœ È"
5 5 3
2 3
13. 14.
period œ1 periodœ41
15. 16.
periodœ2 periodœ4
17. 18.
periodœ6 periodœ1
19. 20.
periodœ2 1 periodœ21
21. 22.
periodœ2 1 periodœ21
23. periodœ 1#, symmetric about the origin 24. periodœ1, symmetric about the origin
25. periodœ4, symmetric about the y-axis 26. periodœ4 , symmetric about the origin1
27. (a) Cos x and sec x are positive in QI and QIV and negative in QII and QIII. Sec x is undefined when cos x is 0. The range of sec x is (_ß "ß _1] [ );
the range of cos x is ["ß1].
(b) Sin x and csc x are positive in QI and QII and negative in QIII and QIV. Csc x is undefined when sin x is 0. The range of csc x is (_ß ß _1] [1 );
the range of sin x is ["ß "].
28. Since cot xœtan x" , cot x is undefined when tan xœ0 and is zero when tan x is undefined. As tan x approaches zero through positive values, cot x approaches infinity.
Also, cot x approaches negative infinity as tan x approaches zero through negative values.
29. D: _ _x ; R: yœ 1, 0, 1 30. D: _ _x ; R: yœ 1, 0, 1
31. cos xˆ 1#‰œcos x cosˆ1#‰sin x sinˆ1#‰œ(cos x)(0)(sin x)( 1) œsin x 32. cos xˆ 1#‰œcos x cosˆ ‰1# sin x sinˆ ‰1# œ(cos x)(0)(sin x)(1)œ sin x 33. sin xˆ 1#‰œsin x cosˆ ‰1# cos x sinˆ ‰1# œ(sin x)(0)(cos x)(1)œcos x
34. sin xˆ 1#‰œsin x cosˆ1#‰cos x sinˆ1#‰œ(sin x)(0)(cos x)( 1) œ cos x
35. cos (AB) œcos (A ( B))œcos A cos ( B) sin A sin ( B) œcos A cos Bsin A ( sin B) cos A cos B sin A sin B
œ
36. sin (AB) œsin (A ( B))œsin A cos ( B) cos A sin ( B) œsin A cos Bcos A ( sin B) sin A cos B cos A sin B
œ
37. If BœA, A œB 0 Ê cos (AB)œcos 0œ1. Also cos (AB)œcos (AA)œcos A cos Asin A sin A cos A sin A. Therefore, cos A sin A 1.
œ # # # # œ
38. If Bœ2 , then cos (A1 2 )1 œcos A cos 21sin A sin 21œ(cos A)(1)(sin A)(0)œcos A and sin (A2 )1 œsin A cos 21cos A sin 21œ(sin A)(1)(cos A)(0)œsin A. The result agrees with the fact that the cosine and sine functions have period 2 .1
39. cos (1x)œcos cos 1 B sin sin x1 œ ( 1)(cos x)(0)(sin x)œ cos x
40. sin (21x)œsin 2 cos ( x)1 cos (2 ) sin ( x)1 œ(0)(cos ( x)) (1)(sin ( x)) œ sin x 41. sinˆ3#1x‰œsinˆ ‰3#1 cos ( x) cosˆ ‰3#1 sin ( x) œ ( 1)(cos x)(0)(sin ( x)) œ cos x 42. cosˆ3#1 x‰œcosˆ ‰3#1 cos xsinˆ ‰3#1 sin xœ(0)(cos x) ( 1)(sin x)œsin x
43. sin 711# œsinˆ4131‰œsin 14 cos 13 cos 14 sin 31 œŠÈ#2‹ˆ ‰#" ŠÈ#2‹ ŠÈ#3‹œ È64È2
44. cos 111#1 œcosˆ41231‰œcos 14 cos 231sin 41sin 231 œŠÈ#2‹ˆ#"‰ŠÈ#2‹ ŠÈ#3‹œ È24È6
45. cos 121 cos 13 14 cos 13 cos 14 sin 13 sin 14 2 3 2 1 3
2 2
œ ˆ ‰œ ˆ ‰ ˆ ‰œˆ ‰ Š ‹ Š ‹ Š"# È# È# È# ‹œ ÈÈ
46. sin 51 sin 23 4 sin 23 cos 4 cos 23 sin 4 3 2 2 1 3
2 2
1 1 1 1 1 1 1
# œ ˆ ‰œ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰œŠÈ# ‹ ŠÈ# ‹ ˆ #"‰ŠÈ# ‹œ ÈÈ
47. cos # 18 œ1 cos #ˆ ‰281 œ1#È#2 œ24È2 48. cos # 11# œ 1 cos #ˆ ‰211# œ 1#È#3 œ 24È3
49. sin # 11# œ 1 cos #ˆ ‰211# œ1#È#3 œ24È3 50. sin # 18 œ 1 cos #ˆ ‰281 œ 1#È#2 œ 24È2 51. tan (AB)œ sin (A B)cos (A B) œ cos A cos B sin A sin B œ
sin A cos B cos A cos B
sin A cos Bcos A sin B
cos A cos B cos A cos B cos A cos B sin A sin B
cos A cos Bcos A cos B
œ tan A tan B 1 tan A tan B
52. tan (AB)œ sin (A B)cos (A B) œ cos A cos B sin A sin B œ
sin A cos B cos A cos B
sin A cos Bcos A sin B
cos A cos B cos A cos B cos A cos B sin A sin B
cos A cos Bcos A cos B
œ tan A tan B 1 tan A tan B
53. According to the figure in the text, we have the following: By the law of cosines, c#œa#b#2ab cos ) 1 1 2 cos (A B) 2 2 cos (A B). By distance formula, c (cos A cos B) (sin A sin B)
œ # # œ #œ # #
cos A 2 cos A cos B cos B sin A 2 sin A sin B sin B 2 2(cos A cos B sin A sin B). Thus
œ # # # # œ
c#œ 2 2 cos (AB)œ 2 2(cos A cos Bsin A sin B) Ê cos (AB)œcos A cos Bsin A sin B.
54. (a) cos Aa Bbœcos A cos B sin A sin B sin )œcosˆ1# )‰ and cos )œsinˆ1# )‰
Let )œ A B
sin Aa Bbœcos’1# aABb“œcos’ˆ1#A‰B“œcos ˆ1# A cos B ‰ sin ˆ1# A sin B‰ sin A cos B cos A sin B
œ
(b) cos Aa Bbœcos A cos B sin A sin B
cos Aa a Bbbœcos A cos aB b sin A sin aBb
cos A B cos A cos B sin A sin B cos A cos B sin A sin B Ê a bœ a b a bœ a b
cos A cos B sin A sin B
œ
Because the cosine function is even and the sine functions is odd.
55. c#œa#b#2ab cos Cœ2#3#2(2)(3) cos (60°)œ 4 9 12 cos (60°)œ1312ˆ ‰"# œ7.
Thus, cœÈ7¸2.65.
56. c#œa#b#2ab cos Cœ2#3#2(2)(3) cos (40°)œ1312 cos (40°). Thus, cœÈ1312 cos 40°¸1.951.
57. From the figures in the text, we see that sin Bœ hc. If C is an acute angle, then sin Cœhb. On the other hand, if C is obtuse (as in the figure on the right), then sin Cœsin (1C)œ hb. Thus, in either case,
hœb sin Cœc sin B Ê ahœab sin Cœac sin B.
By the law of cosines, cos Cœa#2abb# c# and cos Bœa# 2acc# b#. Moreover, since the sum of the
interior angles of a triangle is , we have sin A1 œsin (1(BC))œsin (BC)œsin B cos Ccos B sin C
2a b c c b ah bc sin A.
œˆ ‰hc ’a#2abb# c#“ ’ a# 2acc# b#“ˆ ‰hb œˆ2abch ‰a # # # # #bœ ahbc Ê œ
Combining our results we have ahœab sin C, ahœac sin B, and ahœbc sin A. Dividing by abc gives
h sin A sin C sin B .
bc œðóóóóóóóñóóóóóóóòa œ c œ b
law of sines
58. By the law of sines, sin A# œ sin B3 œÈc3/2. By Exercise 55 we know that cœÈ7.
Thus sin Bœ 3 3 ¶0.982.
2 7 È È
59. From the figure at the right and the law of cosines, b# œa#2#2(2a) cos B
a 4 4a a 2a 4.
œ # ˆ ‰"# œ #
Applying the law of sines to the figure, sin Aa œ sin Bb b a. Thus, combining results, Ê Èa2/2 œÈb3/2 Ê œÉ3#
a#2a œ4 b#œ #3a # Ê 0œ#"a#2a4
0 a 4a 8. From the quadratic formula and the fact that a 0, we have
Ê œ #
aœ 4 È4##4(1)( 8) œ 4È#3 4 ¶1.464.
60. (a) The graphs of yœsin x and yœx nearly coincide when x is near the origin (when the calculator is in radians mode).
(b) In degree mode, when x is near zero degrees the sine of x is much closer to zero than x itself. The curves look like intersecting straight lines near the origin when the calculator is in degree mode.
61. Aœ2, Bœ2 , C1 œ 1, Dœ 1
62. Aœ "#, Bœ2, Cœ1, Dœ"#
63. Aœ 21, Bœ4, Cœ0, Dœ1"
64. Aœ 2L1, BœL, Cœ0, Dœ0
65. (a) amplitudeœk kA œ37 (b) periodœk kB œ365
(c) right horizontal shiftœ œC 101 (d) upward vertical shiftœ œD 25 66. (a) It is highest when the value of the sine is 1 at f(101)œ37 sin (0)25œ62° F.
The lowest mean daily temp is 37( 1) 25œ 12° F.
(b) The average of the highest and lowest mean daily temperaturesœ62° ( 12)°# œ25° F.
The average of the sine function is its horizontal axis, yœ25.
67-70. Example CAS commands:
Maple
f := x -> A*sin((2*Pi/B)*(x-C))+D1;
A:=3; C:=0; D1:=0;
f_list := [seq( f(x), B=[1,3,2*Pi,5*Pi] )];
plot( f_list, x=-4*Pi..4*Pi, scaling=constrained, color=[red,blue,green,cyan], linestyle=[1,3,4,7], legend=["B=1","B=3","B=2*Pi","B=3*Pi"], title="#67 (Section 1.6)" );
Mathematica
Clear[a, b, c, d, f, x]
f[x_]:=a Sin[2 /b (x1 c)] + d
Plot[f[x]/.{aÄ3, bÄ1, cÄ0, dÄ0}, {x, 4 , 4 }] 1 1 67. (a) The graph stretches horizontally.
(b) The period remains the same: periodœ l lB . The graph has a horizontal shift of period."#
68. (a) The graph is shifted right C units.
(b) The graph is shifted left C units.
(c) A shift of „one period will produce no apparent shift. C l l œ ' 69. The graph shifts upwards D units for Dl l ! and down D units for Dl l !Þ
70. (a) The graph stretches A units.l l
(b) For A !, the graph is inverted.