เพิ่มเติมฟังชันตรีโกณมิติe8

5 1 289 5) arctan 1 tan 1 22 2 2 tan 1 cot 2 0 2 2 1 cot2 cosec2 cosec2 1 22 5 cosec 5 0 2 cosec arctan 1 = cosec 2 =5 6) arccos 3 cos 3 0 3 3 cos 3 2 3 cos2 sin2 1 sin2 1 cos2 1 32 2 33 sin 6 2 3 cot arccos 3 = cot 3 = cos sin 3 =3 6 3 = 2 2 7) arccos 1 cos 1 0 3 3 cos 1 02 3

290 5 1 cos2 sin2 1 sin2 1 cos2 1 1 2 8 39 sin 22 0 2 3 = cosec cosec arccos 1 3 =1 sin =1 22 3 = 32 4 8) arccos a cos a 0 cos a 0 a1 0 2 cos2 sin2 1 1 a2 sin2 1 cos2 sin 1 a2 02 sin 2arccos a = sin 2 = 2sin cos = 2 1 a2a = 2a 1 a2 9) cos 5 sin 011 22 arctan 1 tan 1 2, 2 4 tan 4 1 arctan 1 4 arctan cos 5 sin 2 = arctan 1 = 2 4

5 1 291 10) sin 3 2 4 2 arccos 2 cos 2 2 2 0, cos 2 arccos1 2 arccos sin 3 = arccos 2 = 4 2 11) tan 5 1 4 arccos1 cos 1 0, 0 cos 0 1 arccos1 0 arccos tan 5 = arccos1 = 0 4 12) sin 3 cos 2 32 42 arccos sin 3 = arccos 3 5 2 6 arcsin cos 4 = arcsin 2 4 2 tan arccos sin 3 arcsin cos 4 = tan 5 4 6 tan 5 tan 4 6 = 5 6 1 tan tan 4 3 1 3 = 3 1 3 =2 3

292 5 1 21. 1) arcsin 4 sin 4 2 2 5 5 2 sin 4 0 2 5 2 2 cos2 sin2 1 2 cos2 1 sin2 1 4 2 9 5 25 cos 3 0 5 arccos 12 cos 12 0 13 13 cos 12 0 13 2 cos2 sin2 1 sin2 1 cos2 1 12 2 25 13 169 sin 5 0 13 arcsin 16 sin 16 2 65 65 sin 16 0 65 2 cos2 sin2 1 cos2 1 sin2 1 16 2 3969 65 4225 cos 63 0 65 sin = sin = sin cos cos sin = sin cos cos sin cos cos cos sin sin sin sin cos cos cos sin sin sin = sin cos cos cos 3 12 16 4 5 16 sin 5 13 65 5 13 65 = 4 12 63 3 5 63 5 13 65 5 13 65

5 1 293 =1 0 3 2 2 arcsin 4 arccos12 arcsin 16 2 5 13 65 2 2) arctan 1 tan 1 2 3) 3 3 22. 1) tan 2 = 2 tan 1 tan2 = 2 1 2 1 3 1 3 = 3 4 2 = arctan 3 4 2arctan 1 = arctan 3 34 arcsin x sin x 22 cos2 sin2 1 1 x2 cos2 1 sin2 cos 1 x2 22 sin 2arcsin x = sin 2 = 2sin cos = 2x 1 x2 x 1,1 arctan x tan x 22 arctan y 22 tan y tan = tan tan 1 tan tan

294 5 1 =x y 1 xy = arctan x y 1 xy arctan x arctan y = arctan x y 1 xy 2) arctan x tan x 22 arctan y tan y 22 22 2 0 2 tan = tan = tan = tan tan 1 tan tan =x y 1 xy = arctan x y 1 xy = arctan x y 1 xy arctan x arctan y = arctan x y 1 xy 3) arctan x tan x 22 arctan y tan y 22 0 22 tan = tan

5 1 295 = tan tan 1 tan tan =x y 1 xy = arctan x y 1 xy = arctan x y 1 xy arctan x arctan y = arctan x y 1 xy 23. 1) arctan 2 arctan 3 2 3 1, 2 0 arctan 2 arctan 3 = arctan 1 23 23 = arctan 1 = 4 =3 4 arctan1 arctan 2 arctan 3 = 4 3 4 2) arctan 1 arctan 1 2 3 11 1 23 arctan 1 arctan 1 = arctan 11 2 3 1 23 = arctan1 11 23 =4 arctan1 arctan 1 arctan 1 = 4 4 2 2 3

296 5 1 3) arctan 3 arctan 3 4) 33 1 24. 1) arctan 3 arctan 3 = arctan 33 2) 3) 1 33 4) 5) = arctan 0 =0 arctan 3 arctan 0 arctan 3 = 0 arctan 2 arctan 3 2 3 1, 2 0 arctan 2 arctan 3 = arctan 1 2 3 = 2 3 arctan1 =4 =3 4 arctan 1 arctan 2 arctan 3 = 4 3 sec tan 4 1 tan2 = 1 tan2 sec tan = sec2 tan 2 = sec2 = 1 tan2 tan2 =1 tan2 cot2 = sec2 1 cosec2 1 = sec2 1 cosec2 1 = sec2 cosec2 sec sec sin2 = sec 1 sin2 = cos2 cos = cos 2sin cos cos = cos 2 sin 1 1 sin sin2 cos2 1 cos2 sin sin 2

5 1 297 = cos 2 sin 1 sin 2 sin sin 2 = cos 2sin 1 2sin2 sin = cos 2 sin 1 sin 2 sin 1 = cos sin = cot 6) cot cos sin = cos cos sin sin = cos2 sin sin = cos2 sin2 sin =1 sin = cosec 7) 1 sin cos = 1 sin 1 sin cos2 cos 1 sin cos 1 sin = 1 2 sin sin2 cos2 cos 1 sin = 2 2 sin cos 1 sin = 2 1 sin cos 1 sin = 2sec 8) 1 sin = 1 sin 1 sin 1 sin 1 sin 1 sin =1 2sin sin2 1 sin2 = 1 2sin sin2 cos2 = 1 2 sin sin 2 cos2 cos2 cos2

298 5 1 = sec2 2sec tan tan 2 = sec tan 2 9) cos sin = cos sin 1 tan 1 cot 1 sin 1 cos cos sin = cos sin cos sin sin cos cos sin = cos2 sin 2 cos sin cos sin = cos sin cos sin cos sin = sin cos 1 10) cot tan = cot cot 1 tan 1 cot 1 cot 1 1 cot = cot2 1 cot cot 1 cot 1 cot2 1 = cot cot 1 = cot3 1 cot cot 1 cot 1 cot2 cot 1 = cot cot 1 = cot2 cot 1 cot = cot 1 tan = 1 tan cot 11) cot 2 tan = cos 2 sin sin 2 cos = cos2 sin2 sin 2sin cos cos

5 1 299 = cos sin sin 2sin 2cos cos = cos sin 2 sin 2 cos = cos2 sin2 2sin cos = 1 sin 2 = cosec2 12) tan tan = tan 1 tan tan = tan 13) cot tan = cos sin sin cos = cos2 sin2 sin cos 2 cos2 sin2 = 2sin cos = 2cos 2 sin 2 = 2cot 2 14) sin 2 sin 1 = 2sin cos sin cos 2 cos 2cos2 1 cos 1 = 2sin cos sin 2cos2 cos = sin 2 cos 1 cos 2 cos 1 = sin cos = tan

300 5 1 15) sin sin 3 sin 5 = sin 5 sin sin 3 cos cos3 cos5 cos 5 cos cos 3 = 2sin 3 cos 2 sin 3 2cos3 cos 2 cos 3 1 = sin 3 2cos 2 cos 3 2cos 2 1 = sin 3 cos 3 = tan 3 16) cos 20 cos 40 cos80 = 1 2sin 20 cos 20 cos 40 cos80 25. 1) A B C 180 2sin 20 = 1 sin 40 cos 40 cos80 2sin 20 = 1 2sin 40 cos 40 cos80 4sin 20 = 1 sin 80 cos80 4sin 20 = 1 2sin 80 cos80 8sin 20 = 1 sin160 8sin 20 = 1 sin 180 20 8sin 20 = 1 sin 20 8sin 20 =1 8 B C 180 A sin 180 A = sin B C sin A = sin B C 2) A B C 180 B C 180 A cos 180 A = cos B C cos A = cos B C cos A = cos B C

5 1 301 3) A B C 180 A B 180 C sin A sin B sin C = 2sin A B cos A B sin 180 AB 22 = 2sin A 2 B cos A 2 B sin A B = 2sin A 2 B cos A 2 B sin 2 AB = 2sin A B cos A B 2 22 2sin A B cos A B 22 = 2sin A 2 B cos A 2 B cos A 2 B AB AB AB AB = 2sin 180 C 2cos 2 2 cos 2 2 22 2 = 2sin 90 C 2 cos A cos B 2 2 2 = 4cos C cos A cos B 222 = 4cos A cos B cos C 222 4) A B C 180 A B 180 C cos A cos B cosC = 2cos A 2 B cos A 2 B cos 180 A B = 2cos A 2 B cos A 2 B cos A B = 2cos A 2 B cos A 2 B cos 2 AB 2 = 2cos A B cos A B 2cos2 A B 1 22 2 = 2cos A B cos A B 2cos2 A B 1 22 2 = 2cos A 2 B cos A 2 B cos A 2 B 1

302 5 1 = 2 cos 180 C AB AB AB AB 1 2 2sin 2 2 2 sin 2 2 2 = 2cos 90 C 2 sin A sin B 1 2 2 2 = 4 sin C sin A sin B 1 2 2 2 = 1 4sin A sin B sin C 222 26. 1) 4sin2 3 = 0 2) 3) sin2 = 3 4 sin 3 sin 3 2 2 0, 2 sin 3 2 2 3 3 0, 2 sin 34 5 23 3 0, 2 3 , 2 , 4 5 3 3 3 2cos2 3 cos = 0 cos 2cos 3 = 0 cos 0 cos 3 2 0, 2 cos 0 3 2 2 6 0, 2 cos 3 11 2 6 0, 2 6 , 2 , 3 11 2 6 cos2 cos = sin2 cos2 cos = 1 cos2 2cos2 cos 1 2cos 1 cos 1 =0 =0

5 1 303 cos 1 cos 1 2 0, 2 cos 1 3 5 2 3 0, 2 cos 1 0, 2 3, 5 3 4) 3tan2 = 2sec2 1 3tan2 = 2 1 tan2 1 3tan2 = 3 2 tan2 tan2 = 3 tan 3 tan 3 0, 2 tan 3 4 0, 2 tan 3 3 0, 2 32 5 3 3 3 , 2 , 4 5 3 3 3 5) tan = 2sin sin = 2sin cos 0 cos 2sin cos sin = 0 sin 2cos 1 = 0 sin 0 cos 1 2 0, 2 sin 0 0 0, 2 cos 1 3 5 0, 2 2 3 0, 3 , 5 3 6) 3sec cos 2 = 0 3 cos 2 = 0 cos 0 cos 3 cos2 2cos = 0 cos2 2cos 3 = 0

304 51 7) 8) cos 3 cos 1 = 0 27. 1) cos 3 cos 1 1 cos 1 0, 2 cos 3 0 0, 2 cos 1 cosec 5 0, 2 3 3cosec2 2cosec = 0 5 cosec 3cosec 2 = 0 3 cosec 0 cosec 2 5 cosec 1 3 3 0, 2 cosec 0, 2 0, 2 24 33 4 3 cos 2 2cos2 = 1 2 cos 2 2cos2 1 =0 2 =0 2cos2 1 cos =0 2cos2 cos 1 =0 2cos 1 cos 1 1 cos 1 cos 2 0, 2 cos 1 3 5 2 3 0, 2 cos 1 0, 2 3, 2cos2 2cos 2 = 1 2cos2 2 2cos2 1 = 1 6cos2 3 = 0

5 1 305 cos2 =1 cos 2 360 cos 2 360 2 2 2 0 cos 2 45 315 2 225 0 cos 2 135 2 45 ,135 , 225 ,315 2) 4cosec 4sin = 2 2 cot 4 4 sin = 2 2 cos sin 0 sin sin 4 4sin2 = 2 2 cos 4 4 1 cos2 = 2 2 cos 4cos2 2 2 cos = 0 2cos 2cos 2 = 0 cos 0 cos 2 0 360 0 360 2 cos 0 90 270 315 cos 2 45 2 45 ,90 , 270 ,315 3) cos 4sin sin 2 = 2 cos 4sin 2sin cos = 2 cos 2 2sin cos 4sin = 0 cos 2 2sin cos 2 = 0 cos 2 1 2sin = 0 cos 2 sin 1 2 1 cos 1 0 360 cos 2 0 360 sin 1 30 150 2 30 ,150

306 5 1 4) sin 3 cos cos3 sin = cos 28. 1) sin 3 = cos 2) sin 2 cos 2sin cos cos =0 cos 2sin 1 =0 =0 cos 0 sin 1 2 0 360 cos 0 90 270 0 360 150 sin 1 30 2 30 ,90 ,150 , 270 tan2 3 = 0 tan2 = 3 tan 3 tan 3 0, 2 tan 3 4 33 0, 2 tan 3 2 5 33 n3 n3 n cos 2 = sin 1 2sin2 = sin 2sin2 sin 1 = 0 2sin 1 sin 1 = 0 sin 1 sin 1 2 0, 2 sin 1 6 5 2 6 0, 2 sin 1 3 2 2n 6 ,2n 5 2n 3 6 2 n

5 1 307 29. 1) A B C 180 120 35 25 2) A = 180 B C 180 sin A = sin B ab sin 25 = sin120 15 b b = 15sin120 sin 25 15 0.8660 sin A 0.4226 a 30.7383 sin 25 = sin C 15 c c = sin 35 c = 15sin 35 sin 25 15 0.5736 0.4226 20.3597 b 30.7383, c 20.3597 A = 25 A B C 180 C = 180 A B 180 102 41 37 sin C = sin A ca sin 37 = sin102 52.8 a a = 52.8sin102 sin 37 52.8 0.9781 sin C 0.6018 c 85.8154 = sin B b

308 5 1 sin 37 = sin 41 52.8 b b = 52.8sin 41 sin 37 52.8 0.6561 0.6018 57.5641 a 85.8154, b 57.5641 C = 37 3) a2 b2 c2 2bc cos A 12 = 3 2 22 2 3 2 cos A cos A = 3 2 A = 30 b2 a2 c2 2ac cos B 2 cos B 3 2 = 12 22 2 1 cos B = 1 180 30 60 90 2 B= 60 AB C = 180 C 180 AB A = 30 , B = 60 C = 90 4) sin B sin C bc sin 60 = sin C 32 3 3 3 3 sin 60 sin C = 32 = 62 4 C = 75 105 C = 75 A = 180 B C 180 60 75 45

5 1 309 sin B sin A ba sin 60 = sin 45 32 a a = 3 2 sin 45 sin 60 = 23 C = 105 A = 180 B C 180 60 105 15 sin B sin A ba sin 60 = sin15 32 a a = 3 2 sin15 sin 60 =3 3 A = 45 , C = 75 a= 2 3 A = 15 , C = 105 a= 3 3 5) sin B sin C bc sin 45 = sin C 8 12 sin C = 12 sin 45 8 = 3 2 C = 60 120 C = 60 A = 180 B C 180 45 60 75 sin B sin A ba sin 45 = sin 75 8 a a = 8 sin 75 sin 45 =6 2 C = 120 A = 180 B C 180 45 120 15

310 5 1 sin B sin A ba sin 45 = sin15 8a a = 8 sin15 sin 45 =6 2 A = 75 , C = 60 a= 6 2 A = 15 , C = 120 a= 6 2 6) a2 b2 c2 2bc cos A 32 = 42 52 2 4 5 cos A cos A = 4 5 A 36.87 b2 a2 c2 2ac cos B 42 = 32 52 2 3 5 cos B cos B = 3 5 B 53.13 c2 a2 b2 2ab cos C 52 = 32 42 2 3 4 cos C cosC = 0 C = 90 A 36.87 , B 53.13 C = 90

5 1 311 30. AC ABˆC = 90 BAˆ C 55 ACˆB 90 55 35 10 AC = 20 sin ABˆC = sin BAˆ C AC BC sin 90 = sin 55 20 BC BC = 20sin 55 sin 90 16.38 sin ABˆC = sin ACˆB AC AB sin 90 = sin 35 20 AB AB = 20sin 35 sin 90 11.47 = – 16.38 11.47 = 10 2

312 51 31. 126.12 126.12 BCD BD2 = BC2 CD2 2 BC CD cos BCˆD = 402 202 2 40 20 cos30 614.36 BD 24.79 AB = AD , BAˆ D 90 BD2 = AB2 AD2 614.36 = 2 AD2 AD2 = 614.36 2 AD 17.53 ABCD = ABD + BCD = 1 17.53 17.53 1 40 20 sin 30 2 2 353.65 353.65

5 1 313 32. a2 = b2 c2 2bc cos A cos A = b2 c2 a2 2bc cos2 A = b2 c2 a2 2 2bc 2 sin2 A = 1 cos2 A =1 b2 c2 a2 2 2bc 2 2bc 2 b2 c2 a2 2 = 2bc 2 2bc b2 c2 a2 2bc b2 c2 a2 = 2bc 2 a2 b c 2 b c 2 a2 = 2bc 2 = a b c a b cbc a b c a ---------- 1 2bc 2 s=a b c a b c = 2s 2 a b c a b c 2b 2s 2b a b c a b c 2c 2s 2c b c a a b c 2a 2s 2a

314 5 1 1 sin2 A = 2s 2s 2a 2s 2b 2s 2c 2bc 2 = 16s s asb s c 4 bc 2 = 4s s asb s c bc 2 sin A = 2 s s a s b s c bc ABC = 1 bc sin A 2 ABC = 1 bc 2 ss as b s c 2 bc = ss a s b s c a, b c ss a s b s c 33. s= a b c 2 AB AB 60 DAˆ C 45 DBˆA 30 ACD

5 1 315 tan 45 = CD CA CD = CA tan 45 BCD tan 30 = CD CA 60 CD = CA 60 tan 30 CA tan 45 = CA 60 tan 30 CA = CA 60 1 3 CA = 1 CA 60 33 1 1 CA = 60 33 3 1 CA = 60 33 CA = 60 3 3 31 = 60 31 = 60 3 1 31 31 60 3 1 = 2 = 30 3 1 30 3 1

316 51 34. AB2 = AC2 BC2 2 AC BC cos ACˆB = 3.22 2.42 2 3.2 2.4 cos 75 12.02 AB 3.47 3.47

5 1 317 35. CD2 = 1.852 0.42 3.85 CD 1.89 DF 2 = 0.32 0.62 0.45 DF 0.67 CF 2 = 0.72 1.252 2.05 CF 1.43 CF 2 = CD2 DF 2 2 CD DF cos CDˆ F 2.05 3.58 0.45 2 1.89 0.67 cosCDˆ F cosCDˆ F 0.78 CDˆ F 38.74 CD DF 38.74

318 51 36. A 100 100 B BAˆ C 32 DAˆ B 47 32 15 135 32 BDˆ E 90 ADˆ C 43 77 13 AB ADˆ B 30 ADˆ B 43 13 30 CD ABˆD 180 15 30 DAˆ C 47 EBˆD 77 DAˆ C 47 DAˆ B 15 ABD sin ABˆD = sin ADˆ B AD AB sin135 = sin 30 AD 100 AD = 100sin135 sin 30 = 100 2 1 2 2

5 1 319 = 100 2 ACD sin CAˆ D = CD AD CD = AD sin CAˆ D = 100 2 sin 47 103.43 103.43 37. A B AB 500 CD ACD tan 65 = CD AC AC = CD tan 65

320 51 38. BCD tan 35 = CD CB CB = CD tan 35 ABC BAˆ C 90 CB2 = AC 2 AB2 CD 2 CD 2 tan 35 = tan 65 5002 5002 = CD 2 CD 2 tan 35 tan 65 5002 = 1 1 CD2 0.702 2.142 5002 = 1.82CD2 CD 65 370.62 370.62 CD DAˆ C 30 DBˆC 40 DBˆA 180 40 140 DBˆC 40 CDˆ B 90 40 50

5 1 321 DAˆ C 30 DBˆA 140 BDˆ A 180 30 140 10 ABD = sin ABˆD sin ADˆ B AD AB = sin140 sin10 AD 100 = 100sin140 AD sin10 sin ADˆ B = 370.17 AB sin10 = sin DAˆ B 100 BD BD = sin 30 BD = 100sin 30 sin10 = 287.94 BCD sin CBˆD = CD BD CD = BD sin CBˆD = 287.93sin 40 = 185.08 A 370.17 B 287.94 185.08 39. 1) 35 tan 35 0.35 100 arctan 0.35 0.3367 19.29 19.29

322 5 1 2) 20 35 d tan = 20 d 0.35 = 20 d d = 20 0.35 57.14 50,000 1 57.14 1.14 40. 1) s = h cot 1.14 2) s = h cos s cot s sin 45 1 = hsin 90 sin s = h cot h=1 15 30 60 75 90 3 0 23 3 3 23

5 1 323 3) 12.00 15 sin W 90 4) L 41. 1) W=5 L = 6.5 sin 5 6.5 50.28 50.28 L = 1.8 2) sin W L W = 1.4 sin 1.4 1.8 51.06 51.06

324 51 42. A AB AB = 324 1.8 322.2 CD = 1252 1252 125 2 CB = 125 2 2 ABC tan = AB BC = 322.22 125 2 2 3.65 74.68 74.68

5 1 325 43. ABC A AD BC 4 cos ADˆ C 90 2 8sin 2 1) cos 2 = AD AC cos 2 = AD 4 AD = 4cos 2 2) sin = DC 2 AC sin 2 = DC 4 DC = 4sin 2

326 5 1 3) =1 AD BC 44. 2 = 1 4cos 8sin 22 2 = 8 2sin 2 cos 2 = 8sin 8sin AC2 = AB2 BC 2 = 122 52 = 169 AC = 13 EC2 = AE2 AC 2 = 62 132 = 205 EC = 205 ACE AE2 = AC2 EC2 2 AC EC cos ACˆE 62 = 132 205 2 2 13 205 cos ACˆE cos ACˆE 0.9080 ACˆE 24.77 HDF HF 2 = DH 2 DF 2 2 DH DF cos HDˆ F HF = AC = 13 DF = EC = 205 132 = 62 205 2 2 6 205 cos HDˆ F cos HDˆ F 0.4191 HDˆ F 65.22 CH 2 = HD2 DC 2 = 62 122 = 180

5 1 327 CH = 180 ECH EH 2 = EC2 HC2 2 EC HC cos ECˆH 52 = 205 2 180 2 2 205 180 cos ECˆH cos ECˆH 0.9370 ECˆH 20.45 ACE HDF ECH 24.77 , 65.22 20.45 45. f 50 1 cos 2 50 f 50 1 cos 1) 1 cos 1 1 cos 1 0 1 cos 2 0 50 1 cos 100 0 f 100 f 0,100 2) 15 f0 f 0 = 50 1 cos 0 =0 15 0 8 f 90

328 5 1 f 90 = 50 1 cos90 = 50 15 f 180 8 50 f 180 15 100 = 50 1 cos180 8 50 = 100 8 f 270 f 270 = 50 1 cos 270 = 50 3) 25 = 50 1 cos 1 cos = 1 cos 2 = 1 2 0 360 60 300 60 300 25