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Fig. 13.7
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13.12 Show that Arccot 43/32
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Arctan 1/4
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Arccos 12/13.
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Let Arccot 43/32, Arctan 1/4, and Arccos 12/13; then cot 43/32, tan 1/4, and cos 12/13, each angle terminating in the first-quadrant. Taking the tangent of both members of the given relation, we are to show that tan ( ) tan . From Fig. 13.8, tan 5/12. tan (u f) tan u tan f 1 tan u tan f 32>43 1>4 1 (32>43)(1>4) 5 12 tan c
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Fig. 13.8
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13.13 Show that Arctan 1/2
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Arctan 1/8. 1>2 1>5 1 (1>2)(1>5) 1 1 1>8 1>8 7 9 7 9
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We shall show that Arctan 1/2
Arctan 1/5
tan (Arc tan 1>2 and tan (p>4
Arc tan 1>5) Arc tan 1>8)
13.14 Show that 2 Arctan 1/3
Let tan
Arctan 1/7
Arcsec 234>5
Arccsc 217.
1/3,
Arctan 1/3, Arctan 1/7, Arcsec 234>5, and Arccsc 217; then tan 234>5, and csc c 217, each angle terminating in the first quadrant. 1/7, sec l
CHAPTER 13 Inverses of Trigonometric Functions
Taking the tangent of both members of the given relation, we are to show that tan (2 Now tan (2u and, using Fig. 13.9(a) and (b), tan ( tan 2u f) 1 ) tan ( 1 ) 2(1/3) (1/3)2 1 1. 3/4 1
2 tan u tan2 u
tan 2u tan f 1 tan 2u tan f ) 3>5 1>4 (3>5)(1>4)
3/4 1/7 (3/4)(1/7)
Fig. 13.9
13.15 Find the general value of each of the following. (a) Arcsin 22>2 /4 2n , 3 /4 2n (b) Arccos 1/2 /3 2n , 5 /3 2n (c) Arctan 0 2n , (2n 1) 2n where n is a positive or negative integer or zero. 13.16 Show that the general value of (a) Arcsin x (b) Arccos x (c) Arctan x
(a) Let Arcsin x. Then since sin ( (1) 2m )
(d) Arcsin ( 1) (e) Arccos 0 /2 (f) Arctan A 23 B
/2 2n 2n , 3 /2 2n /3 2n , 2 /3
n 2n n
( 1)n Arcsin x Arccos x Arctan x
where n is any positive or negative integer or zero.
sin , all values of Arcsin x are given by and (2) 2m (2m 1) n ( 1)n ; and when ( 1)n . Thus, Arcsin x n 2n and 2n
Now, when n 2m (that is, n is an even integer), (1) may be written as n n 2m 1, (that is, n is an odd integer), (2) may be written as n n ( 1)n Arcsin x, where n is any positive or negative integer or zero. (b) Let or 2n
Arccos x. Then since cos ( ) cos , all values of Arccos x are given by 2n Arccos x, where n is any positive or negative integer or zero.
(c) Let Arctan x. Then since tan ( ) tan , all values of Arctan x are given by 2m and Arctan x, where n is any positive or negative ( ) 2m (2m 1) or, as in (a), by n integer or zero.
13.17 Express the general value of each of the functions of Prob. 13.15, using the form of Prob. 13.16. (a) Arcsin 22>2 n (b) Arccos 1/2 2n (c) Arctan 0 n ( 1)n /4 /3 (d) Arcsin ( 1) n ( 1)n( (e) Arccos 0 2n /2 (f) Arctan A 23 B np p>3 /2)
where n is any positive or negative integer or zero.
CHAPTER 13 Inverses of Trigonometric Functions
SUPPLEMENTARY PROBLEMS
13.18 Write the following in inverse-relation notation. (a) sin Ans. (a) 3/4, Arcsin 3/4, (b) cos (b) 1, Arccos ( 1), (c) tan x (c) x 2, (d) cot (d) 1/2 Arccot 1/2
Arctan ( 2),
13.19 Find the principal value of each of the following. (a) Arcsin 23>2 (b) Arccos A 22>2 B (c) Arctan 1> 23 Ans. (a) /3, (f) 2 /3, (d) Arccot 1 (e) Arcsin ( 1/2) (f) Arccos ( 1/2) (b) 3 /4, (g) /3, (c) (h) /6, /2, (g) Arctan A 23 B (h) Arccot 0 (i) Arcsec A 22 B (d) /4, (i) 3 /4, (e) (j) /6, /2 (j) Arccsc ( 1)
13.20 Evaluate each of the following. (a) sin [Arcsin ( 1/2)] (b) cos A Arccos 23>2 B (c) tan [Arctan ( 1)] (d) sin C Arc cos A 23>2 B D (e) tan (Arcsin 0) Ans. (a) (f) sin (Arccos 4/5) (g) cos [Arcsin ( 12/13)] (h) sin (Arctan 2) (i) Arccos (sin 220 ) (j) Arcsin [cos ( 105 )] (k) Arctan (cot 230 ) (l) Arccot (tan 100 ) (m) sin (2 Arcsin 2/3) (n) cos (2 Arcsin 3/5) (o) sin A 1 Arccos 4>5 B 2 2> 25>5, 210>10
1/2, (b) 23>2, (c) 1, (d) 1/2, (e) 0, (f) 3/5, (g) 5/13, (h) 2> 25 17p 13p p 2p (i) (j) , , (k) , (l) , (m) 4 25>9, (n) 7/25, (o) 1> 210 18 12 9 18
13.21 Show that (a) sin aArc sin (b) cos aArccos (c) sin aArcsin (d) tan aArctan 13.22 Show that 1 2 4 (b) Arc sin 5 4 (c) Arctan 3 (a) Arctan 1 3 3 Arc tan 4 1 Arctan 7 1 1 (d) 2 Arctan Arctan 3 7 Arctan p 4 p 2 p 4 p 4 (e) Arccos 12 1 Arctan 13 4 3 15 Arcsin (f) Arcsin 5 17 1 (g) Arctan a Arctan a 43 32 13 Arccos 85 p (a 0) 2 Arccot 5 13 15 17 4 Arcsin b 5 Arccos 7 b 25 1 77 36 63 65 297 425 2 26 6 (e) cos aArctan (f) tan aArcsin 4 3 3 5 4 5 4 5 Arcsin Arccos Arccos Arccos 12 b 13 5 b 13 12 b 13 12 b 13 63 65 63 16 253 204 323 325
1 2 3 4
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