Show that sin u and tan 1 u have the same sign. 2 in .NET

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6.8 Show that sin u and tan 1 u have the same sign. 2
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(a) Suppose u n 180 . If n is even (including zero), say 2m, then sin (2m 180 ) 1 The case when n is odd is excluded since then tan 2 u is not defined. tan (m 180 ) 0.
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(b) Suppose u n 180 , where 0 < < 180 . If n is even, including zero, u is in quadrant I or quadrant 1 1 II and sin u is positive while 2 u is in quadrant I or quadrant III and tan 2 u is positive. If n is odd, u is in quadrant III or IV and sin u is negative while 1 u is in quadrant II or IV and tan 1 u is negative. 2 2
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6.9 Find all positive values of u less than 360 for which sin u
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1 2.
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There will be two angles (see Chap. 2), one in the third quadrant and one in the fourth quadrant. The reference 1 angle of each has its sine equal to 180 30 210 and 2 and is 30 . Thus the required angles are 360 (NOTE: 210 30 330 . for which sin u
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To obtain all values of 330
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add n 360 to each of the above solutions; thus u
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n 360 and
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n 360 , where n is any integer.)
6.10 Find all positive values of less than 360 for which cos
There are two solutions, 25 in the first quadrant and
1 4u
360 25 335 in the fourth quadrant.
6.11 Find all positive values of
less than 360 , given sin
180 40 140 . But 1440 . Hence, for u we
The two positive angles less than 360 for which sin 0.6428 are u 40 and 1 if 4 u is to include all values less than 360 , must include all values less than 4 360 take the two angles above and all coterminal angles less than 1440 ; that is,
CHAPTER 6 Reduction to Functions of Acute Angles
u and
40 , 400 , 760 , 1120 ; 140 , 500 , 860 , 1220
1 4u
10 , 100 , 190 , 280 ; 35 , 125 , 215 , 305
SUPPLEMENTARY PROBLEMS
6.12 Express each of the following in terms of functions of a positive acute angle.
(a) sin 145 (b) cos 215 (c) tan 440 (d) cot 155 (e) sec 325 (f) csc 190 (g) sin ( 200 ) (h) cos ( 760 ) (i) tan ( 1385 ) (g) sin 20 (h) cos 40 (i) tan 55 (j) cot 70 (k) sec 85 (l) csc 75 (j) cot 610 (k) sec 455 (l) csc 825
Ans.
(a) sin 35 (b) cos 35 (c) tan 80 (d) cot 25 (e) sec 35 (f) csc 10
Find the exact values of the sine, cosine, and tangent of (a) 150 , Ans. (b) 225 , 23>2, (c) 300 , 1> 23 (d) 23>3 120 , (e) (d) 210 , 23>2, (f) 315
(a) 1>2, (b) (c)
1>2, 23 1 23 23>3
22>2,
22>2, 1 23
(e) 1>2,
23>2,
23>2, 1>2,
(f) 22>2, 22>2, 1
Use the appropriate tables to verify that the function has the value stated. (a) (b) (c) (d) (e) sin 155 13 cos 104 38 tan 305 24 sin 114 18 cos 166 51 0.4192 0.2526 1.4071 0.9114 0.9738 u < 360 , for which: (b) cos u 1, (c) sin u 0.6180, (d) cos u 0.5125, (e) tan u 1.5301 (f) (g) (h) (i) ( j) tan 129.48 sin 110.32 cos 262.35 tan 211.84 cos 314.92 1.2140 0.9378 0.1332 0.6210 0.7061
6.15 Find all angles, 0
(a) sin u Ans. 22/2,
(a) 45 , 135 (b) 180 (c) 218 10 , 321 50 or 218.17 , 321.83
(d) 59 10 , 300 50 or 59.17 , 300.83 (e) 123 10 , 303 10 or 123.17 , 303.17
Variations and Graphs of the Trigonometric Functions
7.1 Line Representations of Trigonometric Functions
Let be any given angle in standard position. (See Fig. 7.1 for in each of the quadrants.) With the vertex O as center, describe a circle of radius one unit cutting the initial side OX of at A, the positive y axis at B, and the terminal side of at P. Draw MP perpendicular to OX; draw also the tangents to the circle at A and B meeting the terminal side of or its extension through O in the points Q and R, respectively.
Fig. 7.1
CHAPTER 7 Graphs of the Trigonometric Functions
In each of the parts of Fig. 7.1, the right triangles OMP, OAQ, and OBR are similar, and MP OP OM OP MP OM OM MP OP OM OP MP BR OB OQ OA OR OB
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