เพิ่มเติมฟังชันตรีโกณมิติe8

1| 15 50 6. (Double – angle identities) sin 2 = 2sin cos 7. cos 2 = cos2 sin2 8. sin 1 cos 2 sin =2 cos cos = 1 cos 2 2 9. tan 2 = 2 tan 1 tan2 (Half – angle identities) sin 2 = 1 cos 2 cos 2 = 1 cos 2 tan 2 = 1 cos 1 cos (Product identities) sin = 2sin cos sin = 2cos sin cos = 2cos cos cos = 2sin sin (Factoring identities) sin sin = 2sin cos 22 sin sin = 2cos 2 sin 2 cos cos = 2cos cos 22 cos cos = 2sin cos 22 AB

1| 51 51 sin AC sinus cos OC complement sin tan AE A (tangent ) cot AF complement tan sec OE cosec OF ( ) OA A complement sec y cos x xy 51 y cos x y sin x cos x sin 2 x sin sin cos cos sin cos x = sin 2 x

1| 15 52 cos x = sin cos x cos sin x ----- (1) 2 2 cos 2 sin x = 0 sin x = 0 sin x = cos 2 sin x (1) cos x = sin 2 cos x cos 2 sin x = sin cos x cos sin x 22 = sin x ----- (2) 2 2 y cos x y sin x X2 arccosecant, arcsecant, arccotangent 1. arccosecant arccosecant x, y x cosec y 2y2 2y2 y0 y arccosec x

1| 51 53 2. arcsecant arcsecant x, y x sec y 0y y2 0y y y arcsec x 2 3. arccotangent arccotangent x, y x cot y 0y 0y y arccot x

1| 51 54 1.7 1 51 1. sin cos 1.2 sin3 cos3 2. ABC AC BP 4 1 BP AC P ACB 3. 3 4cos2 a bsin ab ab 1 2sin x 4. 0x2 sin 2x cos 2x sin x cos x 1 0 5. ABCD AB, BC, CD DA 2, 3, 4 6 AC 23 4 6

1| 55 51 6. x, y x2 xy y2 7. 6 4 14:00 1. sin cos = 1.2 ----- (1) sin cos 2 = 1.22 sin2 2sin cos cos2 = 1.44 ----- (2) 1 2sin cos = 1.44 2sin cos = 0.44 sin cos = 0.22 sin cos 3 = 1.23 sin3 3sin2 cos 3sin cos2 cos3 = 1.728 sin3 3sin cos sin cos cos3 = 1.728 (1) (2) sin3 3 0.22 1.2 cos3 = 1.728 sin3 0.792 cos3 = 1.728 sin3 cos3 = 0.936 2. ABC = 1 AC BP 2 1 4 1 = 1 AC sin AC cos 2 2 2= 1 4 sin 4 cos 2 2 = 8sin cos 1 = 4sin cos 1 = 2 2sin cos 2sin cos = 1 sin 2 2 = 1 2

1| 51 56 sin 30o 1 sin150o 1 2 2 2 30o 2 150o 15o 75o 3. 3 4cos2 3 4 1 sin2 = 1 2sin 1 2sin = 3 4 4sin2 1 2sin = 1 4sin2 1 2sin = 1 2sin 1 2sin 1 2sin = 1 2sin a1 b 2 4. sin 2x cos 2x sin x cos x 1 = 0 2sin x cos x 2cos2 x 1 sin x cos x 1 = 0 2sin x cos x sin x 2cos2 x 1 cos x 1 = 0 sin x 2cos x 1 cos x 2cos x 1 = 0 2cos x 1 sin x cos x = 0 2cos x 1 = 0 sin x cos x = 0 2cos x = 1 sin x 1 = 0 cos x cos x = 1 tan x = 1 2 x 3 x 7 x 2 x 4 4 4 3 3 x 3 , 7 , 2 4 4 4 3 3 5. ABC ADC x2 = 4 9 12cos B ----- (1) x2 = 36 16 48cos D ----- (2) ABCD D B 180

1| 57 51 cos D = cos 180o B = cos B cos D (2) cos B x2 = 36 16 48 cos B x2 = 36 16 48cos B ----- (3) (1) (3) cos B = 39 cos B (1) 39 60 60 x2 = 4 9 12 39 60 x2 = 104 5 x 104 x 104 5 5 x AC x 104 5 AC 104 5 6. x2 xy y2 x2 xy y2 x2 xy y2 = x2 y2 2xy cos cos = 1 2 = 120o 120o

1| 51 58 14:00 60o 7. d 2 = 62 42 2 6 4 cos 60o d 2 = 36 16 24 d 2 = 28 d 27 d 27 d d0 14:00 27

248 51 2) 3) sin B = sin C bc sin 75 = sin 60 20 c c = 20sin 60 sin 75 17.93 A B C = 180 C = 180 A B = 180 30 65 = 85 sin A = sin C ac sin 30 = sin 85 a 32 a = 32sin 30 sin 85 16.06 A B C = 180 B = 180 A C = 180 105 60 = 15 sin A = sin B ab sin105 = sin15 a4 a = 4 sin 105 sin15 14.93 sin C = sin B c b sin 60 = sin15 c 4 c = 4sin 60 sin15 13.38

5 1 249 3. 1) ABC = 1 ab sin C 2) 2 3) = 1 15 20 sin 65 4. 2 135.95 ABC = 1 bc sin A 2 = 1 80 5.5 sin103.5 2 213.92 ABC = 1 ac sin B 2 = 1 14.1 27.4 sin112 2 179.10 AC BD ABCD ABC ABC 135 AC2 = AB2 BC2 2 AB BC cos135 = 102 52 2 10 5 cos135 195.71 AC 13.99 BCD BCD 360 2 135 45 2 BD2 = CD2 BC2 2 CD BC cos 45

250 BD = 7.37 51 5. = 102 52 2 10 5 cos 45 54.29 7.37 c AB AB AC 30 180 30 75 2 sin A = sin C BC AB sin 30 60 = sin 75 c c = 60sin 75 sin 30 115.91

ABC 5 1 251 AC 115.91 60 2 115.91 = 291.82 6. E ABCD AB 32 BC 24 AC 2 AB2 BC 2 = 322 242 = 1600 AC = 40 EC = EB = 20 BC2 = EB2 EC2 2 EB EC cos 242 202 202 2 20 20 cos cos = 0.28 73.74 73.74

252 51 7. A, B, C D sin 45 = sin 30 BD 50 BD = 50sin 45 sin 30 = 50 2 ABD = 30 45 75 sin 90 = sin 75 50 2 AD AD = 50 2 sin 75 sin 90 68.30 68.30 1.9 1. A B C D

5 1 253 AB 18 BC CD BAˆ C 45 BAˆ D 60 ABC tan BAˆ C = BC AB tan 45 = BC 18 BC = 18 tan 45 = 18 1 = 18 ABD tan BAˆ D = BD AB tan 60 = BD BD 18 = 18 tan 60

254 51 2. = 18 3 CD BD BC 18 3 18 18 3 18 AB2 = 62 42 2 6 4 cos 30 = 36 16 2 6 4 3 2 10.43 AB 3.23 AB 3.23 3. 1) A B AB 500 BC

5 1 255 BAˆ C 24 tan BAˆ C = BC AB tan 24 = BC 500 BC = 500 tan 24 500 0.4452 222.6 222.6 2) D 200 BD 200

256 5 1 BDˆ C = BC tan BD tan = 222.6 tan 200 = 1.113 48.06 48.06 4. A B CD 90 90 90 CAˆ D ADˆ C 90 CBˆD BDˆ C ACD tan CAˆ D = CD AC tan = CD AC CD = AC tan = 100 tan BCD tan BDˆ C = BC CD

5 1 257 tan = BC CD CD = BC tan = 200 tan 100 tan = 200 tan tan2 = 2 tan = 2 0 CD 100 tan 100 2 100 2 5. A B AB = 4 15 1.7 61.7 CD

258 5 1 ACˆB 60 60 30 tan ACˆB = AB BCˆD 90 BC tan 60 = 61.7 BC BC = 61.7 tan 60 = 61.7 3 ADˆ B 30 tan ADˆ B = AB BD tan 30 = 61.7 BD BD = 61.7 tan 30 = 61.7 3 BCD BD2 = BC 2 CD2 61.7 2 CD2 = BD2 BC 2 3 = 61.7 3 2 = 3 1 61.7 2 3 = 8 61.7 2 3 CD = 2 2 61.7 3 100.76

5 1 259 6. A 100.76 B 140 AB 230 AC ECˆD 30 tan ECˆD = DE CD tan 30 = DE 230 DE = 230 tan 30 = 230 1 3 132.79 DE DB y

260 51 7. A tan 30 = BD B y y CD = 140 tan 30 = 140 1 3 = 140 3 242.49 242.49 1 2 CD = h DAˆ C 45 DBˆC 45 tan DAˆ C = CD AC tan 45 =h AC AC = tan h 45

5 1 261 tan DBˆC = CD BC tan 45 =h BC BC = tan h 45 AB = CB CA = h h tan 45 tan 45 = h tan 1 1 45 tan 45 = h 1 tan 45 tan 1 tan 45 tan tan 45 tan tan 45 tan = h1 tan 1 tan 1 tan 1 tan 1 tan 2 1 tan 2 1 tan 1 tan =h = h 1 tan 1 tan 1 tan 1 tan 1 tan2 =h 2 tan 2 1 tan2 = 2h 2 tan 1 tan2 = 2h tan 2 2h tan 2

262 51 8. 1) AOˆ B 360 72 5 5 OA = OB = 5 AB2 = OA2 OB2 2 OA OB cos AOˆB = 52 52 2 5 5 cos 72 34.55 AB 5.88 5 5.88 29.4 2)

5 1 263 540 108 OAˆ B 54 OAˆ C tan OAˆ C = OC tan 54 AC = 5 AC AC = 5 tan 54 5 1.3764 3.63 AB 2 3.63 7.26 5 7.26 36.3 9. A B C BC 60 D

264 5 1 DAˆ C 37 DAˆ B 49 CAˆ B 49 37 12 ABˆD 90 49 41 ABC sin ABˆC = sin BAˆ C AC BC sin 41 = sin12 AC 60 AC = 60sin 41 sin12 189.33 ADC sin DAˆ C = CD AC sin 37 = CD AC CD = AC sin 37 189.33 0.6018 113.94 189.33 113.94 30 10. A B C AB DE 60

5 1 265 BDˆ A 43.71 BEˆA 38.39 ABD tan BDˆ A = AB BD tan 43.71 = AB BC 30 AB = BC 30 tan 43.71 ABE tan BEˆA = AB BE tan 38.39 = AB BC 60 AB = BC 60 tan 38.39 BC 30 tan 43.71 = BC 60 tan 38.39 BC tan 43.71 30 tan 43.71 = BC tan 38.39 60 tan 38.39 tan 43.71 tan 38.39 BC = 60 tan 38.39 30 tan 43.71 BC = 60 tan 38.39 30 tan 43.71 tan 43.71 tan 38.39 BC 60 0.7923 30 0.9560 0.9560 0.7923

266 51 11. BC 115.20 AB 115.20 30 tan 43.71 138.80 138.80 AOB 2 OA OB cos AOˆB 2 7.5 67.5 cos AOˆB AB2 = OA2 OB2 82 = 7.572.5 7.52 112.5cos AOˆB = 48.5 cos AOˆB 0.4311 AOˆB 64.46 AO B AB2 = O A2 O B2 2 O A O B cos AOˆ B 82 = 62 62 2 6 6 cos AOˆ B 72cos AOˆ B = 8 cos AOˆ B 0.1111 AOˆ B 83.62 AOB 64.46 AO B 83.62

5 1 267 1. sin cos tan cosec sec cot 0 1) 53 0 1 1 0 1 2) 11 10 2 1 2 12 3 2 3 3) 11 2 2 3 23 3 4 2 2 3 23 3 3 4) 14 3 1 32 2 3 3 2 2 3 23 3 23 3 3 5) 35 13 2 6 22 3 1 32 6) 11 3 1 3 3 2 2 12 7) 19 1 3 6 2 2 8) 49 22 4 22 2. cos2 sin2 1 cos2 1 sin2 1 60 2 121 61 3721 2 cos 11 61 1) sin 2 = sin = 60 61 2) cosec = 1 = 1 = 1 = 61 sin sin 60 60 61 3) cos 2 = cos = 11 61 60 4) tan 3 = tan = sin = 61 = 60 cos 11 11 61

268 5 1 11 5) cot = cot = cos = 61 = 11 sin 60 60 61 6) sec 2 = sec 2 = sec 2 = sec =1 = 61 cos 11 3. 1 cot2 cosec2 cosec2 = 1 8 2 64 = 289 =1 15 225 225 cosec = 17 02 15 sin = 15 17 cos2 sin2 1 cos2 = 1 sin2 = 1 15 2 = 64 17 289 cos =8 0 17 2 cos cosec 8 17 409 17 15 255 4. cos2 sin2 1 sin2 = 1 cos2 = 1 35 2 144 = 37 1369 sin = 12 2 37 sec 37 tan sin 12 35 cos 35 sec tan 37 12 49 7 35 35 35 5 5. 1 tan2 sec2 sec2 = 1 1 2 = 37 6 36

5 1 269 sec 37 3 2 6 2 cot 6 cos 6 37 37 cot cos 6 6 37 6 37 37 37 37 6. 1) sec 5 cosec 7 cos 4 = 1 1 cos 4 6 3 3 3 cos 5 sin 7 6 3 =1 1 1 2 33 2 2 =1 2 2) sin 7 cos tan 5 = sin 7 cos tan 5 6 3 4 6 3 4 = 1 1 1 2 2 =2 3) sin 2 cos 2 cot 6 tan 2 = sin 2 cos 2 cot 6 tan 3 33 6 3 3 = 3 1 33 2 2 =3 3 4 4) tan2 cot2 11 sec2 23 cosec2 13 12 12 12 12 = tan2 cot2 sec2 cosec2 12 12 12 12 = tan2 sec2 cot2 cosec2 12 12 12 12 =1 1 =2 7. 1 cos 1 1 cos 0 1 1 cos

270 5 1 0 cos 1 1 1 11 cos cos sec 1 sec 1 8. tan 5 0 tan 0 2 tan 5 9. cos 40 = DB AB DB = AB cos 40 6 0.7660 = 4.5960 6 0.6428 = 3.8568 sin 40 = AD 3.8568 1.7320 6.6800 AB 3.8568 2 7.7136 AD = AB sin 40 tan 30 = AD CD CD = AD cot 30 sin 30 = AD AC AC = AD cosec30 BC CD DB 6.6800 4.5960 11.276 BC 11.276 CA 7.7136

5 1 271 10. ABC A BD = DC = 20 cos 70 = BD AB AB = BD = 20 58.4795 cos 70 0.3420 AB AC BC 2 58.4795 40 156.959 156.959

272 51 11. AB = AC = 90 90 0.9659 = 86.931 cos15 = AD AC AD = AC cos15 x AB AD 90 86.931 3.069 x 3.069 12. 1) 2 1 2 1, 1 22 y 1 cos 2

5 1 273 2) 2 1 2 1, 1 22 y 1 sin 4 2 3) 1 2 1 , 1 2 2 y 1 sin 2 2

274 5 1 4) 1 2 1 , 1 2 2 y 1 sin 2 2 5) 4 2 3, 1 y 2sin 1 1 2

5 1 275 6) 4 2 1, 3 y 2 cos 1 1 2 7) 2 1, 3 y 2sin 2 1

276 51 8) 2 1 3, 1 y 2cos 2 13. 1) () 2) 3) () 4) 5) () 6) () 14. 1) () () cosec 3 3 = 1 2 3 sin 2 3 = 1 sin 3 cos 3 cos 3 sin 3 2 2 =1 3 1 2 1 2 0 =2 2) cos165 = cos 180 15 = cos15

5 1 277 = cos 45 30 = cos 45 cos30 sin 45 sin 30 = 2 3 21 2 2 22 = 62 4 3) sin105 = sin 60 45 = sin 60 cos 45 cos 60 sin 45 = 3 2 12 2 2 22 =62 4 4) tan 285 = tan 360 75 = tan 75 = tan 30 45 = tan 30 tan 45 1 tan 30 tan 45 3 1 3 = 3 1 3 = 33 33 =2 3 5) sec = 1 12 cos 12 =1 cos 4 3 =1 cos 4 cos 3 sin 4 sin 3

278 5 1 =1 21 2 3 22 2 2 =6 2 6) cot 17 = cot 5 12 12 = cot 5 12 = 1 tan 5 12 =1 tan 4 6 =1 tan 4 tan 6 1 tan 4 tan 6 =1 3 3 1 1 3 3 = 33 33 = 32 7) cos 3 cos 7 sin 3 sin 7 = cos 3 7 20 20 20 20 20 20 = cos 2 =0

5 1 279 8) tan 74 tan14 = tan 74 14 1 tan 74 tan14 = tan 60 =3 9) sin 12 cos 5 cos 12 sin 5 = sin 5 12 12 12 12 = sin 2 =1 10) cos100 cos5 sin100 sin 5 = cos 100 5 = cos105 = cos 60 45 = cos 60 cos 45 sin 60 sin 45 = 1 2 32 2 2 22 =2 6 4 11) 1 tan15 = tan 45 tan15 1 tan15 1 tan 45 tan15 = tan 45 15 = tan 60 =3 12) 2 sin 11 sin 7 = cos 11 7 cos 11 7 12 12 12 12 12 12 = cos 3 cos 3 2 =1 2 13) cos165 cos 75 = 1 cos 165 75 cos 165 75 2 = 1 cos 240 cos 90 2 =1 cos 60 2 =1 4

280 5 1 14) sin 80 cos 25 sin100 cos35 cos80 sin 25 cos100 sin 35 1 sin105 sin 55 = 1 sin105 sin 55 1 sin135 sin 65 2 2 2 1 sin135 sin 65 2 = sin105 sin135 = sin 180 75 sin 180 45 = sin 75 sin 45 = sin 30 45 sin 45 = sin 30 cos 45 cos30 sin 45 sin 45 = 1 2 32 2 2 2 22 2 = 6 32 4 15) cos135 cos105 cos 75 = cos 90 45 cos 180 75 cos 75 = cos 90 45 cos 75 cos 75 = cos 90 45 = cos90 cos 45 sin 90 sin 45 =0 2 1 2 2 2 =2 2 16) sin 80 sin 40 sin 20 = 2 cos 80 40 sin 80 40 sin 20 2 2 = 2cos 60 sin 20 sin 20 = 2 1 sin 20 sin 20 2 =0 15. 1) cos2 sin2 1 cos2 1 sin2 1 5 2 144 13 169 cos 12 3 13 2 tan 3 sin 3 , cos 1 2 2 2

5 1 281 sin 5 , cos 12 , sin 3 , cos 1 , tan 5 13 13 2 2 12 tan 3 sin = sin cos cos sin =5 1 12 3 13 2 13 2 = 5 12 3 26 cos = cos cos sin sin = 12 1 53 13 2 13 2 = 12 5 3 26 sin = sin cos cos sin = 5 1 12 3 13 2 13 2 = 5 12 3 26 tan = tan tan 1 tan tan 5 3 3 = 12 1 5 12 = 5 12 3 12 5 3 2) sec 17 cos 8 8 17 cos2 sin2 1 sin2 1 cos2 1 8 2 225 17 289 sin 15 3 17 2 cos2 sin2 1

282 5 1 cos2 1 sin2 1 22 1 22 cos 2 3 2 2 sin 15 , cos 8 , sin 2 , cos 2 , tan 15 tan 1 17 17 2 2 8 sin = sin cos cos sin = 15 2 82 17 2 17 2 =72 34 cos = cos cos sin sin =8 2 15 2 17 2 17 2 = 23 2 34 sin = sin cos cos sin = 15 2 82 17 2 17 2 = 23 2 34 tan = tan tan 1 tan tan 15 1 1 = 8 15 8 1 = 23 7 16. sin2 x cos2 x 1 cos2 x 1 sin2 x 1 3 2 16 5 25

5 1 283 cos x 4 0x 2 sin2 x 5 y cos2 x y 1 cos2 x y 1 sin2 x y 1 5 2 144 13 169 cos x y 12 x y 3 13 2 sin x y = sin x cos y cos xsin y 5 = 3 cos y 4 sin y ------------ 1 13 5 5 ------------ 2 cos x y = cos x cos y sin x sin y ------------ 3 ------------ 4 12 = 4 cos y 3 sin y 13 5 5 ------------ 5 ------------ 6 1 15 = 9 cos y 12 sin y 65 25 25 2 48 = 16 cos y 12 sin y 65 25 25 34 cos y = 63 65 1 20 = 12 cos y 16 sin y 65 25 25 2 36 = 12 cos y 9 sin y 65 25 25 56 sin y = 16 65 sin x 3 , cos x 4 , sin y 16 cos y 63 5 5 65 65 4 1) cot x = cos x = 5 =4 3 3 sin x 5 sin x y 5 cos x y 2) tan x y = = 13 =5 12 12 13 3) sin y = 16 65

284 5 1 4) cos x y = cos x cos y sin xsin y =4 63 3 16 = 204 5 65 5 65 325 17. cot 2x = 1 tan 2x = 1 2 tan x 1 tan2 x = 1 tan2 x 2 tan x 1 12 2 = 21 2 =3 4 18. cos 2 = 1 2sin2 cos 2 0.52 = 1 2sin2 0.52 cos1.04 = 1 2sin2 0.52 0.50 = 1 2sin2 0.52 sin2 0.52 = 0.25 19. 1) sin 0.52 = 0.5 0 0.52 2 sin = sin cos = 0 cos cos sin = sin 1 sin 2) sin 3 = sin 3 cos cos 3 sin 2 2 2 = 1 cos 0 sin = cos 3) tan 90 A = sin 90 A cos 90 A = sin 90 cos A cos90 sin A cos90 cos A sin 90 sin A

5 1 285 = 1 cos A 0 sin A 0 cos A 1 sin A = cos A sin A = cot A 4) sec 90 A = 1 A cos 90 = 1 sin A cos90 cos A sin 90 = 1 0 cos A 1 sin A =1 sin A = cosecA 5) tan 270 A = sin 270 A cos 270 A = sin 270 cos A cos 270 sin A cos 270 cos A sin 270 sin A = 1 cos A 0 sin A 0 cos A 1 sin A 6) sin = cos A cos sin sin sin A cos sin = cot A = sin cos sin cos sin cos cos sin = cos cos cos cos sin cos cos sin cos cos cos cos = tan tan tan tan 7) sin = sin cos cos sin sin cos sin cos = 1 cos sin sin cos

286 5 1 = 1 cot tan 8) cos x 45 cos x 45 = 2cos x 45 x 45 cos x 45 x 45 2 2 = 2cos x cos 45 = 2cos x 2 2 = 2 cos x 9) sin x y sin x y = sin x cos y cos xsin y sin x cos y cos xsin y = sin2 x cos2 y cos2 x sin2 y = sin2 x 1 sin2 y 1 sin2 x sin2 y = sin2 x sin2 x sin2 y sin2 y sin2 x sin2 y = sin2 x sin2 y 10) 1 cos 2 1 1 2sin2 = 2cos sin 2 2cos 2sin cos 2 1 sin2 = 2cos 1 sin = 2 1 sin 1 sin 2cos 1 sin = 1 sin cos = sec tan 11) 2 tan = 2 tan 1 tan2 sec2 = 2sin cos2 cos = 2sin cos = sin 2 12) 1 tan2 = 1 tan2 1 tan2 sec2 = cos2 sin 2 cos2 cos2 = cos2 sin2 = cos 2

5 1 287 13) cos 4 = cos 2 2 14) = 2cos2 2 = 2 2cos2 1 20. 1) 12 1 2) = 2 4cos4 4cos2 1 1 = 8cos4 8 cos 2 21 = 8cos4 8 cos 2 1 = cot2 cosec2 2 tan 2 1 tan 2 = cot 1 cot 1 2 tan =1 11 1 tan2 tan tan 1 2 tan tan 1 tan 1 = 1 tan 1 tan 2 tan tan tan 1 tan 1 tan =2 tan = 2cot arctan 3 tan 3 3 3 , tan 3 22 6 6 3 arctan 3 6 3 arccos 3 cos 3 2 2 5 0, 6 cos 5 3 arccos 3 5 6 2 6 2 cos arccos 3 cos 5 3 2 6 2

288 5 1 3) arccos 1 cos 1 0 3 3 cos2 sin2 1 sin2 1 cos2 1 1 2 8 39 sin 22 0 3 tan arccos 1 = tan 3 = sin cos 22 = 3 1 3 = 22 4) arcsin 2 sin 2 22 3 3 cos2 sin2 1 22 7 cos2 1 sin2 1 39 cos 7 22 3 cot arcsin 2 = cot 3 = cos sin 7 =3 2 3 = 14 2