Reduction to Functions of Acute Angles in .NET

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CHAPTER 6 Reduction to Functions of Acute Angles
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Except for those cases in which a function is not defined, the above relations are also valid when is a quadrantal angle. This may be verified by making use of the fact that 0 and 0 , 90 and 270 , 180 and 180 , and 270 and 90 are coterminal. For example, sin ( 0 ) sin 0 0 sin 0 , sin ( 90 ) sin 270 1 sin 90 , cos ( 180 ) cos 180 , and cot ( 270 ) cot 90 0 cot 270 .
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6.2 Verify the equality of the trigonometric functions for u and its reference angle R where x > 0, y > 0, and r 2x2 y2.
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(a) u is in quadrant I. See Fig. 6.2(a). sin u cos u tan u (b) u is in quadrant II. See Fig. 6.2(b). sin u cos u tan u y r r x sin R x QrR y QxR cos R tan R cot u secu cscu x y r x r y x QyR r QxR csc R cot R sec R y r x r y x sin R cos R tan R cot u secu csc u x y r x r y cot R sec R csc R
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CHAPTER 6 Reduction to Functions of Acute Angles
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(c) u is in quadrant III. See Fig. 6.2(c). sin u cos u tan u y r x y x y x y QrR x QrR tan R cot u sec u csc u x y r x r y x y
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sin R
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cot R r QxR r QyR
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cos R
sec R
csc R
(d) u is in quadrant IV. See Fig. 6.2(d). y r x r y x y QrR cos R y QxR tan R x y r x r y x QyR sec R r QyR csc R
sin u cos u tan u
sin R
cot u sec u csc u
cot R
CHAPTER 6 Reduction to Functions of Acute Angles
6.3 Express the following as functions of positive acute angles. (a) sin 130 , (g) sin 670 ,
(a) sin 130 (b) tan 325 (c) sin 200 (d) cos 370 (e) tan 165 (f) sec 250 (g) sin 670 (h) cot 930 (i) csc 865
(b) tan 325 , (h) cot 930 ,
sin (180 tan (360 sin (200 cos (10 tan (180 sec (250 sin (310 sin 50 cot [210 cot 30 csc [145 csc 35
(c) sin 200 , (i) csc 865 ,
sin 50 tan 35 sin 20 cos 10 tan 15 sec 70 sin 310 cot 210 csc 145 sin (180 180 ) 360 ) cos 40
(d) cos 370 , (e) tan 165 , (f) sec 250 , ( j) sin ( 100 ), (k) cos ( 680 ), (l) tan ( 290 )
130 ) 325 ) 180 ) 360 ) 165 ) 180 ) 360 ) 2(360 )] 2(360 )] [
sin (360 cot (210 csc (180 100 )]
310 ) 180 ) 145 ) sin 80 or sin ( 100 ) sin (260 360 )
(j) sin ( 100 )
sin 100 sin 260 sin 80
sin (260 cos (320 320 ) 2(360 )] 360 )
(k) cos ( 680 ) or cos ( 680 ) (l) tan ( 290 ) or tan ( 290 )
cos 680 cos (360 cos [40 tan 290 tan (70
cos 320
cos 40 290 )] tan 70 tan 70
[ tan (360
6.4 Find the exact value of the sine, cosine, and tangent of (a) 120 , (b) 210 , (c) 315 , (d)
135 ,
120 60 . 1 2
240 ,
(a) 120 is in quadrant II; reference angle sin 120 sin 60 23 2 cos 120
cos 60 210 180 cos 30 360 315 cos 45
tan 120
tan 60
(b) 210 is in quadrant III; reference angle sin 210 sin 30 1 2 cos 210
30 . 23 2 45 . 22 2 tan 315 tan 45 1 tan 210 tan 30 23 3
(c) 315 is in quadrant IV; reference angle sin 315 (d) sin 45 22 2 135
cos 315 360
135 is coterminal with 225 180 45 . sin ( 135 )
225 ; 225 is in quadrant III; reference angle 22 2 tan 45 22 2
sin 45 tan ( 135 )
cos ( 135 ) 1
cos 45
240 is coterminal with
120 ; 120 is in quadrant II; reference angle
sin ( 240 )
sin 60
23 2
cos ( 240 ) tan 60 23
cos 60
tan ( 240 )
CHAPTER 6 Reduction to Functions of Acute Angles
330 is coterminal with sin ( 330 ) sin 30 1 2
30 ; 30 is in quadrant I; reference angle cos 30 23 2 tan ( 330 )
30 . tan 30 23 3
cos ( 330 )
6.5 Use Table 1 (see Appendix 2) to find:
(a) sin 125 14 (b) cos 169 40 (c) tan 200 23 (d) cot 250 44 (e) cos 313 18 (f) sin 341 52 sin (180 cos (180 tan (200 23 cot (250 44 cos (360 sin (360 125 14 ) 169 40 ) 180 ) 180 ) 313 18 ) 341 52 ) sin 54 46 cos 10 20 tan 20 23 cot 70 44 cos 46 42 sin 18 8 0.8168 0.9838 0.3716 0.3495 0.6858 0.3112
6.6 Use Table 2 (see Appendix 2) to find:
(a) tan 97.2 (b) cos 147.8 (c) cot 241.28 (d) sin 194.37 (e) cos 273.1 (f) tan 321.61 tan (180 cos (180 cot (241.28 sin (194.37 cos (360 tan (360 97.2 ) 147.8 ) 180 ) 180 ) 273.1 ) 321.61 ) tan 82.8 cos 32.2 cot 61.28 sin 14.37 cos 86.9 tan 38.39 7.9158 0.8462 0.5480 0.2482 0.7923 0.0541
6.7 Use a calculator to find:
(a) sin 158 38 (b) cos 264 21 (c) tan 288 14 (d) tan 112.68 (e) sin 223.27 (f) cos 314.59 sin (158 cos (264 tan (288 2.39292 0.685437 0.702029 38/60) 21/60) 14/60) 0.364355 0.098451 3.03556
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