Chapter 7
668
Analytic Trigonometry
Section 7.4
■
Sum and Difference Formulas
You should know the sum and difference formulas.
sinsu ± vd 5 sin u cos v ± cos u sin v
cossu ± vd 5 cos u cos v 7 sin u sin v
tan u ± tan v
17 tan u tan v
You should be able to use these formulas to find the values of the trigonometric functions of angles whose sums or
differences are special angles.
tansu ± vd 5
■
■
You should be able to use these formulas to solve trigonometric equations.
Vocabulary Check
1. sin u cos v 2 cos u sin v
3.
2. cos u cos v 2 sin u sin v
tan u 1 tan v
1 2 tan u tan v
4. sin u cos v 1 cos u sin v
5. cos u cos v 1 sin u sin v
6.
1. (a) coss1208 1 458d 5 cos 1208 cos 458 2 sin 1208 sin 458
5
2 !2 2 !6
4
5
!
!
!
1 !2 21 1 !2
(b) cos 1208 1 cos 458 5 2 1
5
2
2
2
3. (a) cos
1 4 1 3 2 5 cos 4 cos 3 2 sin 4 sin 3
p
p
p
5
5
(b) cos
5. (a) sin
(b) sin
!2
2
p
1
?22
p
!2
2
?
p
7p
2
4. (a) sin
14
!2 2 !6
7p
p
1 !3 21 2 !3
2 sin 5 2 2
5
6
3
2
2
2
!
1
(b) sin
!
!
!6 1 !2
4
!2
2
!3
2
2
5
!2 2 !3
2
2
5p
3p
5p
3p
5p
5 sin
cos
1 cos
sin
6
4
6
4
6
1 22 212 23 2 1 12 22 21122
!
52
4
2
3p
5
2
p
5p
p 1
5 sin
5 sin 5
3
6
6
2
1 22 21 232 2 12 22 21122
(b) sin 1358 2 cos 308 5
!3
p
p !2 1 !2 1 1
1 cos 5
1 5
4
3
2
2
2
16
2. (a) sins1358 2 308d 5 sin 1358 cos 308 2 cos 1358 sin 308
1 1221 22 2 2 1 23 21 22 2
5 2
5
tan u 2 tan v
1 1 tan u tan v
!
!
!6 1 !2
4
5p !2 1 !2 1 1
3p
1 sin
5
1 5
4
6
2
2
2
6. (a) sins3158 2 608d 5 sin 3158 cos 608 2 cos 3158 sin 608
5
2 !2
2
52
(b) sin 3158 2 sin 608 5
!2
4
1
?22
2
!2
2
?
!3
2
!6
4
2 !2 !3 2 !2 2 !3
2
5
2
2
2
Section 7.4
7. sin 1058 5 sins608 1 458d
8. 1658 5 1358 1 308
5 sin 608 cos 458 1 cos 608 sin 458
5
5
!3
2
!2
4
!2
?
1
2
1
2
?
5 sin 458 cos 308 2 sin 308 cos 458
2
s!3 1 1d
5
5
5 cos 608 cos 458 2 sin 608 sin 458
5
!2
?
2
!2
4
2
!3
2
sin 1658 5 sins1358 1 308d
5 sin 1358 cos 308 1 sin 308 cos 1358
!2
cos 1058 5 coss608 1 458d
1
5
2
Sum and Difference Formulas
?
!2
!2
2
!2
4
5
5
52
tan 608 1 tan 458
1 2 tan 608 tan 458
1 2 !3
2
1
2
?
!2
2
s!3 2 1d
5 2cos 458 cos 308 2 sin 458 sin 308
52
5
2
5 cos 1358 cos 308 2 sin 1358 sin 308
2
s1 2 !3d
!3 1 1
!3
cos 1658 5 coss1358 1 308d
tan 1058 5 tans608 1 458d
5
?
!3 1 1
1 1 !3
?
1 2 !3 1 1 !3
4 1 2!3
5 22 2 !3
22
!2
2
!2
4
!3
?
2
2
!2
1
?2
2
s!3 1 1d
tan 1658 5 tans1358 1 308d
5
tan 1358 1 tan 308
1 2 tan 1358 tan 308
5
2tan 458 1 tan 308
1 1 tan 458 tan 308
21 1
5
11
!3
3
!3
3
5 22 1 !3
tan 1958 5 tans2258 2 308d
9. sin 1958 5 sins2258 2 308d
5 sin 2258 cos 308 2 cos 2258 sin 308
5
tan 2258 2 tan 308
1 1 tan 2258 tan 308
5
tan 458 2 tan 308
1 1 tan 458 tan 308
5 2sin 458 cos 308 1 cos 458 sin 308
52
5
!2
2
!2
4
?
!3
2
!2
1
2
1
?2
s1 2 !3 d
3
1
3 2 32
5
5
3
31
111
3 2
12
!
cos 1958 5 coss2258 2 308d
5 cos 2258 cos 308 1 sin 2258 sin 308
5 2cos 458 cos 308 2 sin 458 sin 308
52
52
!2
2
!2
4
?
!3
2
2
!2
s!3 1 1d
!
2
1
?2
5
!3
!3
12 2 6!3
5 2 2 !3
6
?
3 2 !3
3 2 !3
669
Chapter 7
670
Analytic Trigonometry
10. 2558 5 3008 2 458
11. sin
sin 2558 5 sins3008 2 458d
1
11p
3p p
5 sin
1
12
4
6
5 sin
5 sin 3008 cos 458 2 sin 458 cos 3008
5 2sin 608 cos 458 2 sin 458 cos 608
52
52
!3
?
2
!2
4
s
!2
2
2
!3 1 1
!2
2
5
1
?2
5
d
cos
cos 2558 5 coss3008 2 458d
5 cos 3008 cos 458 1 sin 3008 sin 458
1
2
?
!2
2
!2
5
4
2
!3
2
?
!2
52
tan
tan 2558 5 tans3008 2 458d
5
tan 3008 2 tan 458
1 1 tan 3008 tan 458
5
2tan 608 2 tan 458
1 2 tan 608 tan 458
7p p p
5 1
12
3
4
sin
1
5 sin
5
5
cos
2
?
2
!2
4
!2
2
1
1
5 cos
5
5
1
2
?
!2
4
!2
2
1
?2
2
p
p
p
p
cos
2 sin sin
3
4
3
4
!2
2
2
!3
2
s1 2 !3d
?
2
p
3p
p
3p
cos 2 sin
sin
4
6
4
6
!2
2
?
!3
2
1
11p
3p p
5 tan
1
4
4
6
21 1
2
!2
2
1
?2
52
2
!3
5
!2
2
3
1 2 s21d
!3
3
5
23 1 !3
3 1 !3
5
212 1 6!3
5 22 1 !3
6
1
7p
p p
5 tan
1
12
3
4
3 2 !3
? 3 2 !3
2
p
p
1 tan
3
4
5
p
p
1 2 tan tan
3
4
5
!3 1 1
1 2 !3
5 22 2 !3
s!3 1 1d
7p
p p
5 cos
1
12
3
4
s!3 2 1d
tan
p
p
p
p
cos 1 sin cos
3
4
4
3
!3
2
3p
p
1 tan
4
6
5
3p
p
1 2 tan
tan
4
6
tan
7p
p p
5 sin
1
12
3
4
22
!2 1
tan
2 !3 2 1
5
5 2 1 !3
1 2 !3
12.
1 2
2
1
2
s1 2 !3d
1
!3
11p
3p p
5 cos
1
12
4
6
5 cos
!2
?
2
5 cos 608 cos 458 2 sin 608 sin 458
5
3p
p
3p
p
cos 1 cos
sin
4
6
4
6
!2
4
2
!2
4
s!3 1 1d
Section 7.4
13. sin
1
17p
9p 5p
5 sin
2
12
4
6
5 sin
5
cos
!
!2
4
s!3 1 1d
5
2
5
1
cos 2
p
p
p
p
cos 2 sin cos
6
4
4
6
!2
?
2
!2
!
!2
s1 2 !3 d
5
2
5
1
tan 2
1 2 s2 !3y3d
!3
?
2
!2
4
2
5
12 1 6!3
5 2 1 !3
6
!2
2
3 1 !3
? 3 1 !3
!3
3
21
!3
3
5 22 1 !3
15. 285 5 225 1 60
sin 2858 5 sins2258 1 608d
5 sin 2258 cos 608 1 cos 2258 sin 608
122 2
!2 1
2
1 2 2 5 2 42 s
!2 !3
2
!
!3 1 1
d
cos 2858 5 coss2258 1 608d
5 cos 2258 cos 608 2 sin 2258 sin 608
52
122 2 12 2 21 2 2 5
!2 1
2
!2
1
2
?
s!3 1 1d
1
5
!3
!2
4
s!3 2 1d
tan 2858 5 tans2258 1 608d
5
tan 2258 1 tan 608
1 1 !3
5
1 2 tan 2258 tan 608 1 2 !3
1 1 !3
5
4 1 2!3
5 22 2 !3 5 2 s2 1 !3d
22
? 1 1 !3
!3
2
2
1
p
p p
5 tan
2
12
6
4
11
52
?
2
p
p
2 tan
6
4
5
p
p
1 1 tan tan
6
4
1 1 s2 !3y3d
3 1 !3
3 2 !3
2
p
p
p
p
cos 1 sin sin
6
4
6
4
tan
5
!2
1
5 cos
tans9py4d 2 tans5py6d
1 1 tans9py4d tans5py6d
2
s1 2 !3 d
4
2
!
4
1
2
2
p
p p
5 cos
2
12
6
4
3
2 1
2
1
2 1 2 2
2 122
1
5
1
!2
17p
9p 5p
5 tan
2
12
4
6
5
2
p
p p
5 sin
2
12
6
4
5 sin
5p
9p
5p
9p
cos
1 sin
sin
4
6
4
6
5 cos
tan
1
sin 2
!
1
5
p
p p
5 2
12
6
4
12 232 2 1 22 21122
2
17p
9p 5p
5 cos
2
12
4
6
5
14. 2
9p
5p
9p
5p
cos
2 cos
sin
4
6
4
6
!2
52
2
Sum and Difference Formulas
!2
2
671
672
Chapter 7
Analytic Trigonometry
17. 21658 5 2 s1208 1 458d
16. 2105 5 308 2 1358
sins21658d 5 sinf2 s1208 1 458dg
sins308 2 1358d 5 sin 308 cos 1358 2 cos 308 sin 1358
5 2sins1208 1 458d
5 sin 308s2cos 458d 2 cos 308 sin 458
5
12212 2 2 2 1 2 21 2 2
!2
1
!2
52
4
!3
5 2 fsin 1208 cos 458 1 cos 1208 sin 458g
!2
52
s1 1 !3d
52
coss308 2 1358d 5 cos 308 cos 1358 1 sin 308 sin 1358
5
1 23 212 22 2 1 11221 22 2
!
!2
4
!
5
11
4
52
52
tan 308 2 s2tan 458d
1 1 tan 308s2tan 458d
3
!2
2
2
1
2
!2
?
2
1
2
?
!2
4
!2
2
2
!3
2
?
!2
2
s1 1 !3d
52
52
tan 1208 1 tan 458
1 2 tan 1208 tan 458
2 !3 1 1
1 2 s2 !3 ds1d
52
1 2 !3
1 1 !3
52
4 2 2!3
22
1 2 !3
!3
?12
5 2 2 !3
18. 158 5 458 2 308
sin 158 5 sins458 2 308d 5 sin 458 cos 308 2 cos 458 sin 308
5
1 22 21 23 2 2 1 22 21122 5
!
!
!
s
!2 !3 2 1
4
d 5 !2s!3 2 1d
4
cos 158 5 coss458 2 308d 5 cos 458 cos 308 1 sin 458 sin 308
5
tan 158 5 tans458 2 308d 5
1 22 21 23 2 1 1 22 21122 5
!
!
s
!2 !3 1 1
4
d 5 !2s!3 1 1d
4
tan 458 2 tan 308
1 1 tan 458 tan 308
!3
12
5
!
3
132
1 1 s1d
!3
4
s!3 2 1d
5 2tans1208 1 tan 458d
5 2 1 !3
!
!2
tans21658d 5 tanf2 s1208 1 458dg
2 s21d
1 33 2s21d
?
5 cos 1208 cos 458 2 sin 1208 sin 458
s1 2 !3 d
!3
2
5 coss1208 1 458d
!
tan 308 2 tan 1358
tans308 2 1358d 5
1 1 tan 308 tan 1358
5
!3
coss21658d 5 cosf2 s1208 1 458dg
5 cos 308s2cos 458d 1 sin 308 sin 458
5
3
3 2 !3
3
3 2 !3
5
5
3 1 !3
3 1 !3
3
3 2 !3
? 3 2 !3 5
12 2 6!3
5 2 2 !3
6
Section 7.4
13p 3p p
5
1
12
4
3
19.
sin
1
3p p
13p
5 sin
1
12
4
3
5
5
!2
!
52
!
5
1 2 !3
1 1 !3
1
52
4 2 2!3
22
2
5 2 2 !3
3p
3p
p
p
cos 2 sin
sin
4
3
4
3
!2
1
?22
2
!2
2
?
!3
2
52
!2
4
s1 1 !3 d
7p
p p
52 2
12
3
4
1
sin 2
2
1
1
1
1 2 12
1 2 12
2
!3
2
21 22 2 2 11221 22 2 5 2 42 s
1
!
!
2
!
1 2 12
!3 1 1
d
1 2 12
7p
p p
p
p
p
p
5 cos 2 2
5 cos 2
cos
1 sin 2
sin
12
3
4
3
4
3
4
5
1
2
7p
p p
p
p
p
p
5 sin 2 2
5 sin 2
cos
2 cos 2
sin
12
3
4
3
4
3
4
5 2
cos 2
2
11221 22 2 1 12 23 21 22 2 5
!
1
!
!
!2
4
s1 2 !3 d
1 p3 2 2 tan1p4 2
2
p
p
1 1 tan12 2 tan1 2
3
4
7p
p p
tan 2
5 tan 2 2
5
12
3
4
5
21 1 !3
1 2 s21ds!3d
52
13p
3p p
5 cos
1
12
4
3
2
134p2 1 tan1p3 2
5
3p
p
1 2 tan1 2 tan1 2
4
3
s1 2 !3 d
!2
4
1
13p
3p p
5 tan
1
12
4
3
tan
12 22 21 23 2
1
?21
2
5 cos
20. 2
2
p
3p
p
3p
cos 1 cos
sin
4
3
4
3
5 sin
cos
tan
Sum and Difference Formulas
tan 2
2 !3 2 1
5 2 1 !3
1 1 s2 !3 ds1d
?
1 2 !3
1 2 !3
673
Chapter 7
674
21. 2
Analytic Trigonometry
1
13p
3p p
52
1
12
4
3
3 14
sin 2
3p
1
p
3
2
24 5 2sin1 4
3p
3
3
3 14
cos 2
3p
1
p
3
3 14
tan 2
3p
1
p
3
3p
1
!2
4
s!3 2 1d
2
122 2 2 1 2 2 5 2 4
!2 1
2
!2 !3
3p
tan
52
1
p
3
1 2 !3
1 1 !3
!2
s!3 1 1d
2
p
3p
1 tan
4
3
1 2 tan
22.
p
3
!3
3p
p
3p
p
cos 2 sin
sin
4
3
4
3
24 5 2tan1 4
5
!2
s1 2 !3 d 5
24 5 cos1 4
5 cos
4
122 1 12 2 2 1 2 24
2
4
52
2
!2 1
!2
52
p
3
3p
p
3p
p
cos 1 cos
sin
4
3
4
3
5 2 sin
52
1
3p
p
tan
4
3
1 2 !3
52
? 1 2 !3 5
21 1 !3
1 2 s2 !3 d
4 2 2!3
5 22 1 !3
22
5p p p
5 1
12
4
6
sin
1 4 1 6 2 5 sin 4 cos 6 1 cos 4 sin 6
p
p
p
5
cos
p
p
p
1 22 21 23 2 1 1 22 21122 5
!
!
!
!2
4
s!3 1 1d
1 4 1 6 2 5 cos 4 cos 6 2 sin 4 sin 6
p
p
p
5
1
2
p p
tan
1
5
4
6
p
p
p
1 22 21 23 2 2 1 22 21122 5
!
!
!
!2
!3
p
p
1 tan
11
4
6
3
5
!3
p
p
1 2 tan tan
1 2 s1d
4
6
3
4
s!3 2 1d
tan
1 2
5 !3 1 2
23. cos 258 cos 158 2 sin 258 sin 158 5 coss258 1 158d 5 cos 408
24. sin 1408 cos 508 1 cos 1408 sin 508 5 sins1408 1 508d 5 sin 1908
Section 7.4
25.
tan 3258 2 tan 868
5 tans3258 2 868d 5 tan 2398
1 1 tan 3258 tan 868
26.
27. sin 3 cos 1.2 2 cos 3 sin 1.2 5 sins3 2 1.2d 5 sin 1.8
Sum and Difference Formulas
tan 1408 2 tan 608
5 tans1408 2 608d 5 tan 808
1 1 tan 1408 tan 608
28. cos
1
p
p
p
p
p p
cos 2 sin sin 5 cos
1
7
5
7
5
7
5
5 cos
29.
tan 2x 1 tan x
5 tans2x 1 xd 5 tan 3x
1 2 tan 2x tan x
2
1
5
5 coss2458d 5
!3
p
p
p
p
p
p
cos 1 cos
sin 5 sin
1
12
4
12
4
12
4
5 sin
12p
35
32. cos 158 cos 608 1 sin 158 sin 608 5 coss158 2 608d
5 sin 3008
33. sin
2
34. cos
1
3p
3p
3p
p
p
p
cos
2 sin
sin
5 cos
1
16
16
16
16
16
16
p
3
5 cos
p !2
5
4
2
!3
2
154p2 2 tan112p 2
5p
p
2 2
5 tan1
36.
5p
p
4
12
1 1 tan1 2 tan1 2
4
12
tan
tan 258 1 tan 1108
5 tans258 1 1108d
35.
1 2 tan 258 tan 1108
5 tan 1358
5 21
5 tan
176p2
5 tan
1p6 2 5
For Exercises 37– 44, we have:
5
5
sin u 5 13
, u in Quadrant II ⇒ cos u 5 2 12
13 , tan u 5 2 12
cos v 5 2 35, v in Quadrant II ⇒ sin v 5 45, tan v 5 2 43,
y
y
(−3, 4)
(−12, 5)
5
13
2
30. cos 3x cos 2y 1 sin 3x sin 2y 5 coss3x 2 2yd
31. sin 3308 cos 308 2 cos 3308 sin 308 5 sins3308 2 308d
52
675
u
x
v
x
Figures for Exercises 37– 44
!3
3
2
!2
2
676
Chapter 7
Analytic Trigonometry
37. sinsu 1 vd 5 sin u cos v 1 cos u sin v
5
38. cossu 2 vd 5 cos u cos v 1 sin u sin v
1
113212 52 1 12 132152
5
52
3
12
4
5 2
63
65
5
39. cossu 1 vd 5 cos u cos v 2 sin u sin v
1
5 2
5
41. tansu 1 vd 5
12
13
5
4
5
16
65
1 2 tan u tan v
5
5
4
2 12
1 s2 3 d
12s
5
2 12
ds d
2 43
21
5
2 12
1 2 59
42. cscsu 2 vd 5
1 21 2
7
5 2
4
43. secsv 2 ud 5
5
5
36 20 56
1
5
65 65 65
3 5
2 2
145212 12
13 2 1 5 21 13 2
52
tan u 1 tan v
9
63
52
4
16
5
1
1
5
cossv 2 ud cos v cos u 1 sin v sin u
1
4 5
s2 35 ds2 12
13 d 1 s5 ds13 d
5
212 352 1 1135 21452
40. sinsv 2 ud 5 sin v cos u 2 cos v sin u
212 52 2 1132152
3
12
13
44. tansu 1 vd 5
1
1
20 5 56
1
s36
d
s
d
65
65
65
65
56
5
cotsu 1 vd 5
33
48 15
1
52
65 65
65
1
1
5
sinsu 2 vd 2sinsv 2 ud
1
65
5
33
2 s2 33
d
65
s2 125 d 1 s2 43 d
tan u 1 tan v
5
5
1 2 tan u tan v 1 2 s2 12
ds2 43 d
2 74
4
9
52
63
16
1
1
16
5
52
tansu 1 vd 2 63
63
16
For Exercises 45–50, we have:
7
7
sin u 5 2 25
, u in Quadrant III ⇒ cos u 5 2 24
25 , tan u 5 24
cos v 5 2 45, v in Quadrant III ⇒ sin v 5 2 35, tan v 5 34
y
y
v
u
x
x
25
5
(−24, −7)
(− 4, −3)
Figures for Exercises 45– 50
45. cossu 1 vd 5 cos u cos v 2 sin u sin v
46. sinsu 1 vd 5 sin u cos v 1 cos u sin v
4
7
3
5 s2 24
25 ds2 5 d 2 s2 25 ds2 5 d
7
5 s2 25
ds2 45 d 1 s2 2425 ds2 35 d
3
55
28
72
4
5 125
1 125
5 100
125 5 5
Section 7.4
47. tansu 2 vd 5
5
2 34
11s
7
24
48. tansv 2 ud 5
11
ds d
3
4
5
s4 d 2 s24 d
tan v 2 tan u
5
7
1 1 tan v tan u 1 1 s34 ds24
d
3
tan u 2 tan v
1 1 tan u tan v
7
24
Sum and Difference Formulas
2 24
52
39
32
44
117
11
24
39
32
5
1
1 5
53 5
cossu 1 vd
3
5
Use Exercise 45 for cossu 1 vd.
1
50. cossu 2 vd 5 cos u cos v 1 sin u sin v 5 2
5
24
25
212 542 1 12 257 212 352
96
21
117
1
5
125 125 125
51. sinsarcsin x 1 arccos xd 5 sinsarcsin xd cossarccos xd 1 sinsarccos xd cossarcsin xd
5 x ? x 1 !1 2 x2
? !1 2 x2
5 x2 1 1 2 x2
51
1
1
x
θ
x
θ = arcsin x
θ = arccos x
52. Let
u 5 arctan 2x and v 5 arccos x
tan u 5 2x
cos v 5 x.
4x 2 + 1
1
2x
u
1 − x2
v
x
1
sinsarctan 2x 2 arccos xd 5 sinsu 2 vd
5 sin u cos v 2 cos u sin v
5
5
1 − x2
θ
1 − x2
2x
!4x2 1 1
sxd 2
2x2 2 !1 2 x2
!4x2 1 1
1
!4x2 1 1
s!1 2 x2 d
44
117
1
1
117
5 44 5
tansv 2 ud 117
44
cotsv 2 ud 5
49. secsu 1 vd 5
5
7
677
Chapter 7
678
Analytic Trigonometry
53. cossarccos x 1 arcsin xd 5 cossarccos xd cossarcsin xd 2 sinsarccos xd sinsarcsin xd
5 x ? !1 2 x2 2 !1 2 x2
?x
50
(Use the triangles in Exercise 51.)
54. Let
u 5 arccos x
v 5 arctan x
and
cos u 5 x
tan v 5 x.
1
1 + x2
1 − x2
x
u
v
x
1
cossarcos x 2 arctan xd 5 cossu 2 vd
5 cos u cos v 1 sin u sin v
5 sxd
5
1!1 11 x 2 1 s!1 2 x d1!1 x1 x 2
2
2
2
x 1 x!1 2 x2
!1 1 x2
55. sins3p 2 xd 5 sin 3p cos x 2 sin x cos 3p
56. sin
5 s0dscos xd 2 s21dssin xd
1 2 1 x2 5 sin 2 cos x 1 sin x cos 2
p
p
5 s1dscos xd 1 ssin xds0d
5 sin x
57. sin
5 cos x
1 6 1 x2 5 sin 6 cos x 1 cos 6 sin x
p
p
5
p
58. cos
14
5p
2
2 x 5 cos
1
scos x 1 !3 sin xd
2
59. cossp 2 ud 1 sin
52
1 2 1 u2 5 cos p cos u 1 sin p sin u 1 sin 2 cos u 1 cos 2 sin u
p
p
p
5 s21dscos ud 1 s0dssin ud 1 s1dscos ud 1 ssin uds0d
5 2cos u 1 cos u
50
1
2
p
60. tan
2u 5
4
p
2 tan u
1 2 tan u
4
5
p
1 1 tan u
1 1 tan tan u
4
tan
p
5p
5p
cos x 1 sin
sin x
4
4
!2
2
scos x 1 sin xd
Section 7.4
Sum and Difference Formulas
61. cossx 1 yd cossx 2 yd 5 scos x cos y 2 sin x sin ydscos x cos y 1 sin x sin yd
5 cos2 x cos2 y 2 sin2 x sin2 y
5 cos2 xs1 2 sin2 yd 2 sin2 x sin2 y
5 cos2 x 2 cos2 x sin2 y 2 sin2 x sin2 y
5 cos2 x 2 sin2 yscos2 x 1 sin2 xd
5 cos2 x 2 sin2 y
62. sinsx 1 yd sinsx 2 yd 5 ssin x cos y 1 sin y cos xdssin x cos y 2 sin y cos xd
5 sin2 x cos2 y 2 sin2 y cos2 x
5 sin2 xs1 2 sin2 yd 2 sin2 y cos2 x
5 sin2 x 2 sin2 x sin2 y 2 sin2 y cos2 x
5 sin2 x 2 sin2 yssin2 x 1 cos2 xd
5 sin2 x 2 sin2 y
63. sinsx 1 yd 1 sinsx 2 yd 5 sin x cos y 1 cos x sin y 1 sin x cos y 2 cos x sin y
5 2 sin x cos y
64. cossx 1 yd 1 cossx 2 yd 5 cos x cos y 2 sin x sin y 1 cos x cos y 1 sin x sin y
5 2 cos x cos y
65. cos
12
3p
2
2 x 5 cos
3p
3p
cos x 1 sin
sin x
2
2
66. cossp 1 xd 5 cos p cos x 2 sin p sin x
5 s21d cos x 2 s0d sin x
5 s0dscos xd 1 s21dssin xd
5 2cos x
5 2sin x
2
2
− 2p
2p
−6
−2
−2
67. sin
12
3p
2
6
1 u 5 sin
3p
3p
cos u 1 cos
sin u
2
2
68. tansp 1 ud 5
5 s21dscos ud 1 s0dssin ud
5
5 2cos u
tan p 1 tan u
1 2 tan p tan u
0 1 tan u
1 2 s0d tan u
5 tan u
2
3
− 2p
2p
−6
6
−2
−3
679
680
Chapter 7
Analytic Trigonometry
1
sin x 1
69.
sin x cos
2
1
2
p
p
1 sin x 2
51
3
3
p
p
p
p
1 cos x sin 1 sin x cos 2 cos x sin 5 1
3
3
3
3
2 sin xs0.5d 5 1
sin x 5 1
x5
1
sin x 1
70.
sin x cos
2
1
p
2
2
p
p
1
2 sin x 2
5
6
6
2
1
2
p
p
p
p
1
5
1 cos x sin 2 sin x cos 2 cos x sin
6
6
6
6
2
2 cos xs0.5d 5
1
2
cos x 5
1
2
x5
1
cos x 1
71.
cos x cos
2
1
p 5p
,
3 3
2
p
p
2 cos x 2
51
4
4
1
2
p
p
p
p
51
2 sin x sin 2 cos x cos 1 sin x sin
4
4
4
4
1 22 2 5 1
22 sin x
!
2 !2 sin x 5 1
sin x 5 2
sin x 5 2
x5
1
!2
!2
2
5p 7 p
,
4 4
Section 7.4
Sum and Difference Formulas
tansx 1 pd 1 2 sinsx 1 pd 5 0
72.
tan x 1 tan p
1 2ssin x cos p 1 cos x sin pd 5 0
1 2 tan x tan p
tan x 1 0
1 2fsin xs21d 1 cos xs0dg 5 0
1 2 tan xs0d
tan x
2 2 sin x 5 0
1
sin x
5 2 sin x
cos x
sin x 5 2 sin x cos x
sin xs1 2 2 cos xd 5 0
sin x 5 0
cos x 5
or
x 5 0, p
1
Analytically: cos x 1
73.
cos x cos
2
1
x5
1
2
p 5p
,
3 3
2
p
p
1 cos x 2
51
4
4
p
p
p
p
2 sin x sin 1 cos x cos 1 sin x sin 5 1
4
4
4
4
1 22 2 5 1
2 cos x
!
!2 cos x 5 1
1
cos x 5
!2
!2
cos x 5
2
p 7p
,
4 4
x5
2
1
2
1
p
p
Graphically: Graph y1 5 cos x 1
1 cos x 2
4
4
The points of intersection occur at x 5
1
74. tansx 1 pd 2 cos x 1
2 and y
2
5 1.
0
p
7p
and x 5
.
4
4
2
p
50
2
Answers: s0, 0d, s3.14, 0d ⇒ x 5 0, p
−2
4
0
−4
2p
2p
681
682
Chapter 7
75. y 5
Analytic Trigonometry
1
1
sin 2t 1 cos 2t
3
4
1
1
(a) a 5 , b 5 , B 5 2
3
4
C 5 arctan
76.
3
b
5 arctan < 0.6435
a
4
y<
!1132 1 1142 sins2t 1 0.6435d
y5
5
sins2t 1 0.6435d
12
2
(b) Amplitude:
5
feet
12
(c) Frequency:
1
B
2
1
5
5
5 cycle per second
period 2p 2p p
2
y1 5 A cos 2p
1T 2 l2
y2 5 A cos 2p
1T 1 l2
y1 1 y2 5 A cos 2p
t
x
t
x
1T 2 l2 1 A cos 2p 1T 1 l2
3
y1 1 y2 5 A cos 2p
5 2A cos 2p
t
x
t
x
4
3
t
x
t
x
t
x
t
x
cos 2p 1 sin 2p sin 2p
1 A cos 2p cos 2p 2 sin 2p sin 2p
T
l
T
l
T
l
T
l
4
t
x
cos 2p
T
l
78. False.
77. False.
cossu ± vd 5 cos u cos v 7 sin u sin v
sinsu ± vd 5 sin u cos v ± cos u sin v
1
79. False. cos x 2
2
p
p
p
5 cos x cos 1 sin x sin
2
2
2
80. True.
1
5 scos xds0d 1 ssin xds1d
sin x 2
2
1
2
p
p
2 x 5 2cos x
5 2sin
2
2
5 sin x
81. cossnp 1 ud 5 cos np cos u 2 sin np sin u
82. sinsnp 1 ud 5 sin np cos u 1 sin u cos np
5 s21d scos ud 2 s0dssin ud
5 s0dscos ud 1 ssin uds21dn
5 s21dnscos ud, where n is an integer.
5 s21dn ssin ud, where n is an integer.
n
83. C 5 arctan
b
b
a
⇒ sin C 5
, cos C 5
!a2 1 b2
!a2 1 b2
a
1
!a2 1 b2 sinsBu 1 Cd 5 !a2 1 b2 sin Bu
84. C 5 arctan
? !a2 1 b2 1 !a2 1 b2 ? cos Bu2 5 a sin Bu 1 b cos Bu
a
b
a
a
b
⇒ sin C 5
, cos C 5
!a2 1 b2
!a2 1 b2
b
1
!a2 1 b2 cossBu 2 Cd 5 !a2 1 b2 cos Bu
? !a2 1 b2 1 sin Bu ? !a2 1 b2 2
5 b cos Bu 1 a sin Bu
5 a sin Bu 1 b cos Bu
b
a
Section 7.4
85. sin u 1 cos u
86. 3 sin 2u 1 4 cos 2u
a 5 1, b 5 1, B 5 1
(a) C 5 arctan
Sum and Difference Formulas
a 5 3, b 5 4, B 5 2
b
p
5 arctan 1 5
a
4
(a) C 5 arctan
sin u 1 cos u 5 !a2 1 b2 sinsBu 1 Cd
1
p
4
5 !2 sin u 1
b
4
5 arctan < 0.9273
a
3
3 sin 2u 1 4 cos 2u 5 !a2 1 b2 sinsBu 1 Cd
< 5 sins2u 1 0.9273d
2
a
3
5 arctan < 0.6435
b
4
(b) C 5 arctan
a
p
(b) C 5 arctan 5 arctan 1 5
b
4
3 sin 2u 1 4 cos 2u 5 !a2 1 b2 cossBu 2 Cd
sin u 1 cos u 5 !a2 1 b2 cossBu 2 Cd
1
p
5 !2 cos u 2
4
2
88. sin 2u 2 cos 2u
87. 12 sin 3u 1 5 cos 3u
a 5 1, b 5 21, B 5 2
a 5 12, b 5 5, B 5 3
(a) C 5 arctan
< 5 coss2u 2 0.6435d
b
5
5 arctan
< 0.3948
a
12
12 sin 3u 1 5 cos 3u 5 !a2 1 b2 sinsBu 1 Cd
(a) C 5 arctan
sin 2u 2 cos 2u 5 !a2 1 b2 sinsBu 1 Cd
< 13 sins3u 1 0.3948d
(b) C 5 arctan
a
12
5 arctan
< 1.1760
b
5
12 sin 3u 1 5 cos 3u 5
!a2
1
b2
cossBu 2 Cd
< 13 coss3u 2 1.1760d
b
p
5 arctans21d 5 2
a
4
1
5 !2 sin 2u 2
(b) C 5 arctan
sin 2u 2 cos 2u 5 !a2 1 b2 cossBu 2 Cd
1
b p
5
⇒ a50
a
2
!a2 1 b2 5 2 ⇒ b 5 2
B51
1
2 sin u 1
2
p
5 s0dssinud 1 s2dscosud 5 2 cos u
2
90. C 5 arctan
cossx 1 hd 2 cos x cos x cos h 2 sin x sin h 2 cos x
5
h
h
5
cos x cos h 2 cos x 2 sin x sin h
h
5
cos xscos h 2 1d 2 sin x sin h
h
5
cos xscos h 2 1d sin x sin h
2
h
h
p
4
2
a
3p
52
⇒ a 5 b, a < 0, b < 0
b
4
!a2 1 b2 5 5 ⇒ a 5 b 5
25!2
2
B51
1
5 cos u 1
91.
2
a
p
5 arctans21d 5 2
b
4
5 !2 cos 2u 1
89. C 5 arctan
p
4
2
3p
5!2
5!2
52
sin u 2
cos u
4
2
2
683
Chapter 7
684
Analytic Trigonometry
92. (a) The domains of f and g are the sets of real numbers, h Þ 0.
(b)
(c) The graphs are the same.
2
h
0.01
0.02
0.05
0.1
0.2
0.5
f shd
20.504
20.509
20.521
20.542
20.583
20.691
g shd
20.504
20.509
20.521
20.542
20.583
20.691
−3
3
−2
(d) As h → 0, f shd approaches 20.5.
As h → 0, gshd approaches 20.5.
y
93.
m1 5 tan a and m2 5 tan b
b 1 d 5 908 ⇒ d 5 908 2 b
y 1 = m 1x + b 1
a 1 u 1 d 5 908 ⇒ a 1 u 1 s908 2 bd 5 908 ⇒ u 5 b 2 a
θ
δ
Therefore, u 5 arctan m2 2 arctan m1.
β
α
x
For y 5 x and y 5 !3x we have m1 5 1 and m2 5 !3.
y 2 = m 2x + b2
u 5 arctan!3 2 arctan 1
5 608 2 458
5 158
94. For m2 > m1 > 0, the angle u between the lines is:
u 5 arctan
11 1 m m 2
m2 2 m1
1
2
m2 5 1
m1 5
1
!3
1 2
12
u 5 arctan
11
95.
1
!3
1
!3
5 arctans2 2 !3 d 5 158
3
− 2p
2p
1
Conjecture: sin2 u 1
2
1
2
p
p
1 sin2 u 2
51
4
4
−3
1
sin2 u 1
2
1
2 3
p
p
p
p
1 sin2 u 2
5 sin u cos 1 cos u sin
4
4
4
4
2
p
p
2
5
3
5
sin2 u
cos2 u sin2 u
cos2 u
1 sin u cos u 1
1
2 sin u cos u 1
2
2
2
2
sin u cos u
1
!2
!2
4 3
4 1 3sin u cos 4 2 cos u sin 4 4
5 sin2 u 1 cos2 u
51
2
1
sin u cos u
2
!2
!2
4
2
Section 7.4
Sum and Difference Formulas
685
96. (a) To prove the identity for sinsu 1 vd we first need to prove the identityfor cossu 2 vd. Assume
0 < v < u < 2p and locate u, v, and u 2 v on the unit circle.
y
C
u−v
1
B
D
u
−1
A
v
O
x
1
21
The coordinates of the points on the circle are:
A 5 s1, 0d, B 5 scos v, sin vd, C 5 scossu 2 vd, sinsu 2 vdd, and D 5 scos u, sin ud.
Since /DOB 5 /COA, chords AC and BD are equal. By the distance formula we have:
!fcossu 2 vd 2 1g2 1 fsinsu 2 vd 2 0g2 5 !scos u 2 cos vd2 1 ssin u 2 sin vd2
cos2su 2 vd 2 2 cossu 2 vd 1 1 1 sin2su 2 vd 5 cos2 u 2 2 cos u cos v 1 cos2 v 1 sin2 u 2 2 sin u sin v 1 sin2 v
fcos2su 2 vd 1 sin2su 2 vdg 1 1 2 2 cossu 2 vd 5 scos2 u 1 sin2 ud 1 scos2 v 1 sin2 vd 2 2 cos u cos v 2 2 sin u sin v
2 2 2 cossu 2 vd 5 2 2 2 cos u cos v 2 2 sin u sin v
22 cossu 2 vd 5 22scos u cos v 1 sin u sin vd
cossu 2 vd 5 cos u cos v 1 sin u sin v
Now, to prove the identity for sinsu 1 vd, use cofunction identities.
3 2 2 su 1 vd4 5 cos3 1 2 2 u2 2 v4
p
sinsu 1 vd 5 cos
5 cos
p
1p2 2 u2 cos v 1 sin1p2 2 u2 sin v
5 sin u cos v 1 cos u sin v
(b) First, prove cossu 2 vd 5 cos u cos v 1 sin u sin v using the figure containing points
y
1
As1, 0d
u−v
D
C
Bscossu 2 vd, sinsu 2 vdd
u
B
u−v
1
v
A
Cscos v, sin vd
−1
Dscos u, sin ud
on the unit circle.
−1
Since chords AB and CD are each subtended by angle u 2 v, their lengths are equal. Equating
fdsA, Bdg2 5 fdsC, Ddg2 we have scossu 2 vd 2 1d2 1 sin2su 2 vd 5 scos u 2 cos vd2 1 ssin u 2 sin vd2.
Simplifying and solving for cossu 2 vd, we have cossu 2 vd 5 cos u cos v 1 sin u sin v.
Using sin u 5 cos
1p2 2 u2 we have
sinsu 2 vd 5 cos
3 p2 2 su 2 vd4 5 cos3 1p2 2 u2 2 s2vd4
5 cos
1p2 2 u2 coss2vd 1 sin1p2 2 u2 sins2vd
5 sin u cos v 2 cos u sin v.
x
686
97.
Chapter 7
Analytic Trigonometry
f sxd 5 5sx 2 3d
72x
8
72x
y5
8
8y 5 7 2 x
f sxd 5
98.
y 5 5sx 2 3d
y
5x23
5
x 5 7 2 8y ⇒ f 21sxd 5 28x 1 7
y
135x
5
7 2 f21sxd
8
7 2 s28x 1 7d
5
8
5x
f s f 21sxdd 5
x
135y
5
f 21sxd 5
x 1 15
5
f s f 21sxdd 5 f
1
2
3
x 1 15
x 1 15
55
23
5
5
1
4
f 21s f sxdd 5 28
17 28 x2 1 7
5x
2
x 1 15
2 5s3d
55
5
5 x 1 15 2 15
5x
f 21s f sxdd 5 f 21s5sx 2 3dd 5
5
5x 2 15 1 15
5
5
5x
5
5sx 2 3d 1 15
5
5x
99. f sxd 5 x2 2 8
f is not one-to-one so f
100.
21
does not exist.
f sxd 5 !x 2 16, x ≥ 16
y 5 !x 2 16
y 2 5 x 2 16
x 5 y 2 1 16 ⇒ f 21sxd 5 x2 1 16, x ≥ 0
f s f 21sxdd 5 !sx2 1 16d 2 16 5 x
f 21s f sxdd 5 s!x 2 16 d 1 16 5 x
2
2
101. log3 34x23 5 4x 2 3
102. log8 83x 5 3x2
103. elns6x23d 5 6x 2 3
104. 12x 1 eln xsx22d 5 12x 1 xsx 2 2d
5 12x 1 x2 2 2x
5 x2 1 10x
Section 7.5
Section 7.5
■
Multiple-Angle and Product-to-Sum Formulas
Multiple-Angle and Product-to-Sum Formulas
You should know the following double-angle formulas.
(a) sin 2u 5 2 sin u cos u
(b) cos 2u 5 cos2 u 2 sin2 u
(b)
5 2 cos2 u 2 1
(b)
5 1 2 2 sin2 u
(c) tan 2u 5
■
■
2 tan u
1 2 tan2 u
You should be able to reduce the power of a trigonometric function.
(a) sin2 u 5
1 2 cos 2u
2
(b) cos2 u 5
1 1 cos 2u
2
(c) tan2 u 5
1 2 cos 2u
1 1 cos 2u
You should be able to use the half-angle formulas. The signs of sin
u
u
u
and cos depend on the quadrant in which lies.
2
2
2
!1 2 2cos u
1 1 cos u
u
(b) cos 5 ± !
2
2
■
■
(a) sin
u
5±
2
(c) tan
u 1 2 cos u
sin u
5
5
2
sin u
1 1 cos u
You should be able to use the product-sum formulas.
1
(a) sin u sin v 5 fcossu 2 vd 2 cossu 1 vdg
2
1
(b) cos u cos v 5 fcossu 2 vd 1 cossu 1 vdg
2
1
(c) sin u cos v 5 fsinsu 1 vd 1 sinsu 2 vdg
2
1
(d) cos u sin v 5 fsinsu 1 vd 2 sinsu 2 vdg
2
You should be able to use the sum-product formulas.
(a) sin x 1 sin y 5 2 sin
1
(c) cos x 1 cos y 5 2 cos
2 1
x1y
x2y
cos
2
2
1
2 1
2
x1y
x2y
cos
2
2
(b) sin x 2 sin y 5 2 cos
2
1
2 1
x1y
x2y
sin
2
2
(d) cos x 2 cos y 5 22 sin
1
2 1
2
x1y
x2y
sin
2
2
2
687