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13.1 Find the principal value of each of the following. (a) Arcsin 0 (c) Arctan 23 (d) Arccot 23 0 p>3 p>6 (e) Arcsec 2 (f) Arccsc A (g) Arccos 0 (h) Arcsin ( 1) /3 22 B /2 /2 p>4 (i) Arctan ( 1) (j) Arccot 0 (k) Arcsec A (l) Arccsc ( 2) /2 22 B 3p>4 5 /6 /4 (b) Arccos ( 1)
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CHAPTER 13 Inverses of Trigonometric Functions
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13.2 Express the principal value of each of the following to the nearest minute or to the nearest hundredth of a degree. (a) (b) (c) (d) (e) (f) Arcsin 0.3333 Arccos 0.4000 Arctan 1.5000 Arccot 1.1875 Arcsec 1.0324 Arccsc 1.5082 19 28 or 19.47 66 25 or 66.42 56 19 or 56.31 40 6 or 40.10 14 24 or 14.39 41 32 or 41.53 (g) Arcsin ( (h) Arccos ( (i) Arctan ( (j) Arccot ( (k) Arcsec ( (l) Arccsc ( 0.6439) 0.4519) 1.4400) 0.7340) 1.2067) 4.1923) 40 5 or 40.08 116 52 or 116.87 55 13 or 55.22 126 17 or 126.28 145 58 or 145.97 13 48 or 13.80
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13.3 Verify each of the following. (a) sin (Arcsin 1/2) sin /6 1/2 (e) Arccos [cos ( /4)] Arccos 22>2 (b) cos [Arccos ( 1/2)] cos 2 /3 1/2 (f) Arcsin (tan 3 /4) Arcsin ( 1) (c) cos C Arc sin A 22>2 B D cos ( p>4) 22>2 (g) Arccos [tan ( 5 /4)] Arccos ( 1) (d) Arcsin (sin /3) Arcsin 23>2 p>3 p>4 /2
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13.4 Verify each of the following. (a) Arcsin 22>2 (b) Arccos 0 Arcsin 1/2 /2 /4 ( /6 /4) /12 /4 Arctan 1
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Arctan ( 1)
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13.5 Evaluate each of the following: (a) cos (Arcsin 3/5),
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(b) sin [Arccos ( 2/3)],
3/5,
(c) tan [Arcsin ( 3/4)]
Arcsin 3/5; then sin
being a first-quadrant angle. From Fig. 13.2(a), cos 4/5
cos (Arcsin 3/5) (b) Let Arccos ( 2/3); then cos
2/3, being a second-quadrant angle. From Fig. 13.2(b), sin u 25>3
sin [Arccos ( 2/3)] (c) Let Arcsin ( 3/4); then sin 3/4,
being a fourth-quadrant angle. From Fig. 13.2(c), tan u 3> 27 3 27>7
tan [Arcsin ( 3/4)]
Fig. 13.2
CHAPTER 13 Inverses of Trigonometric Functions
Arcsin 4/5).
Arcsin 12/13 Arcsin 4.5 12/13 and sin 4/5, and being first-quadrant angles. From Fig. 13.3(a) and (b), sin ( sin cos ) cos 5 #4 13 5 sin 56 65
13.6 Evaluate sin (Arcsin 12/13
Let and Then sin
sin (Arcsin 12/13
Arcsin 4/5)
12 # 3 13 5
Fig. 13.3
13.7 Evaluate cos (Arctan 15/8
Let and Then tan 15/8 and sin
Arcsin 7/25).
Arctan 15/8 Arcsin 7/25 7/25, and being first-quadrant angles. From Fig. 13.4(a) and (b), cos ( cos cos ) sin 15 # 7 17 25 sin 297 425
cos (Arctan 15/8
Arcsin 7/25)
8 # 24 17 25
Fig. 13.4
CHAPTER 13 Inverses of Trigonometric Functions
13.8 Evaluate sin (2 Arctan 3).
Let Arctan 3; then tan 3, being a first-quadrant angle. From Fig. 13.5, sin 2u 2 sin u cos u 2 A 3> 210 B A 1> 210 B 3>5 sin (2 Arctan 3)
Fig. 13.5
13.9 Show that Arcsin 1> 25
Arc sin 2> 25
p>2.
1> 25 and sin f 2> 25, each angle /2 or, taking the sines of both mem-
Let Arcsin 1> 25 and Arcsin 2> 25; then sin u terminating in the first quadrant. We are to show that bers, that sin ( ) sin /2. From Fig. 13.6(a) and (b), sin ( ) sin cos cos sin
1 # 1 25 25
2 # 2 25 25
sin p>2
Fig. 13.6
13.10 Show that 2 Arctan 1/2
Let Arctan 1/2 and We are to show that 2 Now tan 2u 1
Arctan 4/3.
Arctan 4/3; then tan 2(1>2) (1>2)2 1/2 and tan 4/3. tan . or, taking the tangents of both members, that tan 2 1 4>3 tan f.
2 tan u tan2 u
CHAPTER 13 Inverses of Trigonometric Functions
Arcsin 3/5 Arccos 15/17.
13.11 Show that Arcsin 77/85
Let Arcsin 77/85, Arcsin 3/5, and Arccos 15/17; then sin 77/85, sin 3/5, and cos 15/17, each angle terminating in the first-quadrant. Taking the sine of both members of the given relation, we are ) sin . From Fig. 13.7(a), (b), and (c), to show that sin ( sin (u f) sin u cos f cos u sin f 77 # 4 85 5 36 # 3 85 5 8 17 sin c.
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