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  1. 1117 a) amp 5 per 180 b) amp 2 per 90 c) amp 3 per 72
  2.  
  3. 1118 a) amp 4 per 720 b) amp 1 per 1440 c) amp 0.5 per 1080
  4.  
  5. 1119 a) amp 2 per 360 b) amp 4 per 180 c) amp 1 per 360 (mittlinje på y=3)
  6.  
  7. 1120 a) förskj. 20 grader vänster. b) 50 grader höger
  8.  
  9. 1121 a) 45 grader vänster b) 30 grader höger
  10.  
  11. 1122 a) amp 3 per 180 för 25 höger b) amp 4 per 180 15 höger
  12.  
  13. 1123 a) amp 1.5 per 72 för 12 höger b) amp 2 per 120 för 30 vänster
  14.  
  15. 1124 a) A 1.5 k 2 | A 1 k 2
  16.  
  17. 1125 a) 3 -3 b) 2 0 c) 5 -1
  18.  
  19. 1126 a) y = 1.5sin 2(x-30)
  20.  
  21. 1127 a) y = sin 2x b) y = 5 sin 2(x-15) c) y = 3 sin 0.5(x+40)
  22.  
  23. 1128 a) 2 sin 1(x-45) b) 2.5 sin 3(x-45) c) 2 sin 0.25(x+120)
  24.  
  25. 1129 a) 2 sin 4x + 3
  26.  
  27. 1131 a) b 7 k 0.5
  28.  
  29. 1133 a) y = 4 sin 2(x-30) + 3
  30.  
  31. 1201 a) 10 + n * 360°, 170 + n * 360° b) 40 + n * 360°, 140 + n * 360°
  32.  
  33. 1202 a) ca 45 grader. 45/3 = 15 | 15 + n * 120°, 45 + n * 120° b) ca 40 grader. 40/2=20 | 20 + n * 180°, 70 + n * 180°
  34.  
  35. 1203 a) ca 20 grader | 50 + n * 180°, 190 + n * 360°
  36.  
  37. 1204 a) sin(2x + 10) x = 5 | 5 + n * 180°, 75 + n * 180°
  38.  
  39. 1206 a) -45 grader 360 - 45 = 315 | 315 + n * 360, 225 + n * 360 b) -65 grader 360 - 65 = 295 180 - 65 = 115 2x= 295 2x = 115 x= 148 x= 123
  40.  
  41. 1207 a) 3 sin x = 2.457 | 2.457/3 = 0.819 | arcsin 0.819 = 55° | x = 55° + n * 360°, x = 125° + n * 360° b) 2 sin x + 1 = 0 | 2 sin x = -1 | sin x = -0.5 | -30 grader | x = 330 + n * 360, x= 210 + n * 360
  42.  
  43. 1208 a) arccos 0.819 = 35° | +- 35° + n * 360 b) arccos 0.342 = 70° | +- 70° + n * 360
  44.  
  45. 1209 a) cos 2x = 0.94 | arccos 0.94 = 20° | 2x = 20°, x = 10° | x= +- 10° + n * 180 b) cos 3x = 0.707 | 45 grader | 3x = 45, x = 15 | x= +- 15° + n * 180
  46.  
  47. 1210 a) cos (x - 40) = 0.259 | 75 grader | cos (x - 40) = +- 75 | 35, -35[-35+360=325] | 35 + n * 360, 325 + n * 360
  48.  
  49. 1221 a) sin 0.174 = 10° | Lösningar: 10°, 170°, 370° b) 15 ° | Lösningar 15°, 165°,
  50.  
  51. 1223 a) sin 0.5x = 15° | 0.5x = 15, x = 30 | Lösningar: 30°, 150°, -210°, -330°. b)
  52.  
  53. 1233 a) arctan 0 = 0° + n * 180 b) 2 tan x = -3 tan | -3/2 = - 1.5 = -56° | -56° + 180° = 124° + n * 180° b) arctan -7/3 = -67° | -67 + 180 = 113° + n * 180°
  54.  
  55. 1238 a) sin x = 5 cos x | sin x / cos x = 5 | tan x = 5 | arctan 5 = 78.7° | sin 78.7° = 5 cos 78.7° + n * 180° b) 2 sin x = 3 cos x | arctan 3/2 = 56° + n * 180° c) 5 sin x - cos x = 0 | 5 sin x = cos x | 5 tan x = 1 | tan x = 11.3° + n * 180
  56.  
  57. 1245 a) sin x = sin 60° | 180° - 60° = 120° b) sin x = sin 20° | 180° - 20° = 160°
  58.  
  59. 1246 a) sin 3x = sin 60° | 60/3 = 20° 120/3 = 40° | 20° + n * 120°, 40° + n * 120° b) sin 2x = 30° | 30/2 = 15 150/2= 75 | 15° + n * 180°, 75° + n * 180°
  60.  
  61. 1247 a) sin 3x = sin x | 3x = x + n * 360 | 2x = n * 180 | 3x = 180 - x + n * 180| 3x + x = 180 + n * 360 | x = 45 + n * 180 | x + n * 360, 45 + n * 90°
  62.  
  63. 1248 a) sin(x + 30°) = sin(2x - 20°) | 2x - 20 = x + 30 | x = 50 + n * 360° (x + 30°) = 180° - (2x - 20°) | x + 30° = 180° - 2x + 20° | 3x = 170° + n * 360 | x = 56,7° + n * 120
  64.  
  65. 1249 a) n * 120° b) 3x = x + 30 | 2x = 30 | x = 15 | 15° + n * 180 | 3x = - (x + 30) | 3x = - x - 30 | 4x = -30, x = -7.5(+90=82.5) | 15° + n * 180, 82.5° + n * 180
  66.  
  67. 1250 a) cos 4x = cos 5x, n * 40 b) cos (x - 20°) = cos 3x | 3x = x - 20 | 2x = -20 | x = - 10 | 80° + n * 180 | 3x = - (x - 20) | 3x = - x + 20 | 4x = 20 | x = 5 | 5° + n * 90
  68.  
  69. 1259 a) cos 2x = sin 4x | cos 2x = cos(90° - 4x) | 2x = 90 - 4x | 6x = 90 | x = 15° + n * 60° | 2x = -( 90 - 4x) | 2x = - 90 + 4x | -2x = -90 | 2x = 90 | x = 45° + n * 180° | 15° + n * 60° ,45° + n * 180°
  70.  
  71.  
  72.  
  73.  
  74.  
  75.  
  76.  
  77.  
  78. cos x = sin (90° - x)
  79.  
  80. sin x = cos (90° - x)
  81.  
  82. AVSTÅNDSFORMELN: d = sqrt( (x1 -x2)^2 + (y1 - y2)^2 )
  83.  
  84. COSINUSSATSEN: c^2 = a^2 + b^2 - 2ab * cos C
  85.  
  86. SUBTRAKTIONSSATSEN FÖR COSINUS: cos (u-v) = cos u cos v + sin u sin v
  87.  
  88.  
  89. sin(v+u)=sin v cos u + cos v sin u
  90.  
  91. sin(v−u)=sin v cos u − cos v sin u
  92.  
  93. cos(v+u)=cos v cos u − sin v sin u
  94.  
  95. cos(v−u)=cos v cos u + sin v sin u
  96.  
  97.  
  98. 1301 a) cos (180° + x) = cos 180° cos x - sin 180° sin x | -1 * cos x - 0 sin x = - cos x b) sin (90 - x) = sin 90° cos x + cos 90° sin x | 1 cos x + 0 sin x = cos x
  99.  
  100. 1302 a) sin (x - 180) = sin x cos 180° + cos x sin 180° = sin x -1 + cos x 0 = - sin x
  101.  
  102. 1305 a) cos 85° cos 25° + sin 85° sin 25° = cos (85-25)= cos 60 = 0.5 b) cos (70 + 50) = cos 120° = -0.5
  103.  
  104. 1307 a) cos (60 + x) + cos (60 - x) = (cos 60 cos x - sin 60 sin x) + (cos 60 cos x + sin 60 sin x) = (0.5 cos x - 0.866 sin x) + (0.5 cos x + 0.866 sin x) = cos x
  105.  
  106.  
  107. FORMLER FÖR DUBBLA VINKELN
  108.  
  109. sin 2x = 2 sin x cos x
  110.  
  111. cos 2x = cos^2 x - sin^2 x
  112.  
  113. cos 2x = 2 cos^2 x - 1
  114.  
  115. cos 2x = 1 - 2 sin^2 x
  116.  
  117. 1317 COS x = 0.8
  118. SIN x = 0.6
  119.  
  120. TAN x = 1.33333
  121.  
  122. a) sin 2x = 2 0.6 * 0.8 = 0.96
  123. b) cos 2x = 0.8^2 - 0.6^2 = 0.28
  124. c) c) 0.96/0.28 = 3.43
  125.  
  126. 1318 a) cos x = 3/4 = 0.75 | cos 2x = 2 0.75^2 - 1 = 0.125 = 1/8 b) sin x = 1/2 | cos x = 1 - 2 1/2 ^ 2 = 0.5 = 1/2 c) cos x = -5/6 | cos 2x = 2(-5/6)-1 = 0.388888... = 7/18
  127.  
  128. 1319 sin x = 4/5 = 53°
  129.  
  130. a) cos x = (1 - 0.8)^2 = 0.6
  131. b) tan x = sinx/cosx = 0.8/0.6 = 1.33333333333333
  132. c) sin 2x = 2 sin x cos x = 0.96
  133.  
  134.  
  135. 1326 a) 2 sin x cos x = 0.5 = sin 2x | sin 2x = 0.5 | sin x = 0.25 x = 15° + n * 180° | 75° + n * 180°
  136. b) cos x sin x = 0 | n * 90°
  137.  
  138.  
  139. HAHAHAHAHAHAHHAHAHAHAHA
  140.  
  141. FUNKTIONEN y = a sin x + b cos x
  142.  
  143. a sin x + b cos x = m sin (x + v) NÄR a > 0 och b > 0
  144.  
  145.  
  146. 1336 y = m sin (x + v)
  147.  
  148. a) y = 3 sin x + 4 cos x | a = 3 b = 4 | sqrt(3^2 + 4^2) = sqrt(25)=5 | tan v = 4/3 v = 53° | 5 sin (x + 53.1°)
  149. b) y = 12 sin x - 5 cos x | a = 12 b = 5 | sqrt(12^2 + 5^2) ) sqrt(169) = 13 | tan v = 5/12 v = arctan 5/12 = 22.6°. | 13 sin (x - 22.6°)
  150.  
  151.  
  152. 1339 a) y = sin x + sqrt(3 cos x ) | a = 1 b = sqrt 3 | sqrt(1^2 + 3) = 2 | tan v = 1/sqrt3 v = 30° | 2 sin (x + 30) | AMP 2 för. 60 vänster
  153.  
  154. 1401 a) 90 * pi/180 = pi/2
  155. b) 60 * pi/180 = pi/3
  156. c) 45 * pi/180 = pi/4
  157. d) 300 * pi/180 = pi/
  158.  
  159. 1403 a) 2pi = 360°
  160. b) pi/2 = 90°
  161. c) pi = 180°
  162. d) pi/3= 60°
  163.  
  164. 1404 a) 1,5pi = 270°
  165. b) 5pi = 180*5 = 500+400=900°
  166. c) pi/10 = 18°
  167. d) 4pi/3 = 1.333333pi = 180 * 1.33333333 = 240°
  168.  
  169. 1406 a) 0.873
  170. b) 1.92
  171. c) 0.489
  172. d) 3.70
  173.  
  174. 1407 ° * pi/180 = rad
  175.  
  176. a) 68.8°
  177. b) 43.0°
  178. c) 573°
  179. d) 57.3°
  180.  
  181. 1410 0.5° = 0.009 rad | 380 000 km | 380 000 * 0.009 = 3420 km
  182.  
  183. 1411 sin x = 0.65 x = 0.708 + n * 2pi
  184.  
  185. 1425 a) sin pi/6 = 1/2 b) cos pi/3 = 1/2 c) 1/sqrt2
  186.  
  187. 1426 a) tan pi/3 = sin pi/3/cos pi/3 = (sqrt3/2)/ 1/2 = sqrt3
  188.  
  189. 1429 a) 20*180 = 3600° sin 360 = 0 b) 0 c) 1
  190.  
  191. 1430 a) 0 b) icke-definierad c) 1
  192.  
  193. 1431 a) sin 95pi/6 = -0.5
  194.  
  195. 1432 a) sin x = 0.5 x = pi/6 x = pi - pi/6 = 5pi = 5pi/6
  196.  
  197. 1444 a) 1 år b) 4 timmar c) 8 timmar
  198.  
  199. 1445 a) 150+50*1 = 200°
  200. b) 150+50*-1 = 100°
  201. c) 150+50*sin(0.52(6)=151 grader
  202. d) 175 = 150+50*sin(0.52(t) | 150+50*sin(0.52) = 174.844 | 175 / 174.844 = 1.0009
  203.  
  204. 1446 u(t) sin 314t
  205.  
  206. a) 0
  207. b) 183.1 volt
  208. c) -24.7 volt
  209. d) 220 = 311 sin 314(t) 0.707 = sin 314t | sin t = 0.0023
  210.  
  211.  
  212. DERIVATOR
  213.  
  214. Repetition av derivata
  215.  
  216. 2101 a) (2 + h)^2 + 5 - (2^2) + 5 = 4 + 4h + h^2 - 4 / h = 4 + h
  217. b) 4
  218.  
  219. 2102 a) f(x) = x^2 + 7x
  220. (x + h)^2 + 7(x + h) - (x^2 + 7x) / h = x^2 + 2xh + 7x + 7h - x^2 + 7x = 2xh + 7h / h = 2x + 7
  221.  
  222. 2102 a) den går kraftigt uppåt
  223. b) den går mindre kraftigt uppåt
  224. c) rak
  225. d) nedåt
  226.  
  227.  
  228. 2112 a) 5x^4 + 6x^2
  229. b) 12
  230.  
  231. 2113 a) e^x + 3e^3
  232. b) 1 - 4e^x
  233.  
  234. 2121 a) 1/x^4
  235. b) -6/5x^3
  236.  
  237. 2122 a) 0
  238.  
  239.  
  240. EN SJUHELVETES MASSA DERIVERINGSREGLER
  241.  
  242. Der. Logaritmsfunktioner: f(x) = ln x f´(x) = 1/x | f(x) = lg x f´(x) = 1 / x * ln 10
  243.  
  244. 2232 a) 5 - 1/x
  245. b) 1/x - 2x
  246. c) 5/x
  247.  
  248. 2233 a) (1/2x) - 8
  249. b) 8x + 4/x
  250. c) 1/x + 1/1 = 1/x + 1
  251.  
  252. 2234 a) 1/(x+1)
  253. b) 1/2x * 2 = 2/2x = 1/x
  254. c) 2/3x * 3 = 6/3x= 2/x
  255.  
  256. 2235 a) 5/x ln 10
  257. b) 3 lg 2x = 3 / 2x ln 10 * 2 | 6 / 2x ln 10 | 3 / x ln 10
  258. c) 3 ln x = 3/x
  259.  
  260. 2236 a) 6x - 4 + 1/x | 6 - 4 + 1/1 = 3
  261. b) 6/x - 4 - 4x^3 | 6/1 - 4 - 4(1)^3 = 6-4-4 = -2
  262. c) 10/x - 0 = 10
  263. d) 1/4x - 3x^2 - 2/x = 1/4 - 3^2 - 2/1 = 0.25 - 3 - 2 = -4.75
  264.  
  265.  
  266. BERIVATAN AV BRODUKT!: y = f(x) * g(x) y'= g'(x) + f'(x) * g(x)
  267.  
  268. 2247 a) y = x + e^x | y' = (x * e^x) + (1 * e^x) = xe^x + e^x
  269. b) (x^2 * -e^-x) + (2x * e^-x) = lol
  270.  
  271. 2248 a) (x^2 * cos x) + (2x * sin x)
  272. b) (e^x * -sin x) + (e^x * cos x) = e^x cos x - e^x sin x
  273.  
  274. 2249 a) (x * 2^x ln 2) + (2^x)
  275. b) (x^3 * cos x) + (3x^2 * sin x)
  276.  
  277. 2250 a) (ln x * cos x) + (1/x * sin x)
  278. b) (4e^3x * sin 4x) + (3e^3x * cos 4x)
  279.  
  280.  
  281. BERIVATAN AV EN BVOT!: ( f´(x)*g(x) - f(x)*g'(x) ) / (g(x) )^2
  282. =
  283. 2263 a) f'(x) = 1 g'(x) = 1 | 1 * (x-1) - x * 1) / (x-1)^2 = (x-1) * x / (x-1)^2 = x cacnerf xd
  284. b) f'(x) = -1 g'(x) = 1 | -1 * (x+2) - (2+x) * 1 / (2+x)^2 = -(x+2) - (2+x) / (2+x)^2 | -x-2-2-x = -2x-4 | -2(x+2) / (x+2)^2 | (x^2) | -2/2+x
  285.  
  286. ______________________________________________________________________________________________________________________________________________________
  287.  
  288. KAPITEL TRE!!!!!!!!!!!!!!!!
  289.  
  290. 3119 -cos pi/2 = 0
  291.  
  292. -cos 2(pi/2) = 1
  293.  
  294. -cos 4(pi/2) = -1
  295.  
  296. a) f(sin x) | F(-cos x) | F(-cos (pi/2) = 0 | F(-cos pi/2) + 1
  297. b) f(sin 2x) | F(-cos 2x/2) | F(-cos(2(pi/2)/2)= 0.5 | F(-cos(2(pi/2)/2) + 0.5 = 1
  298. c) f(sin 4x) | F(-cos 2x/2) | F(-cos 2(pi/2)/4) = -0.25 | F(-cos 2(pi/2)/2) + 1.25 = 1
  299.  
  300. 3120 f(x) = e^2x + 2x = 0.5e^2x + 2^2 | F(x)= 0 | 0.5e^2x + 2^2 = 0 | 0.5e^2(0) + 4 = 0.5 | F(0.5e^2(0) + 0^2) + 1.5
  301.  
  302. 3121 sin pi = 0
  303. (sin 2 pi)/2 = 0
  304. (sin pi)/3 = 0
  305.  
  306. a) f(cos x) | F(sin x) | F(sin pi) = 0 | F(sin pi) + 2 = 2
  307. b) f(cos 2x) | F(sin 2x/2) | F(sin 2pi/2) = 0 | F(sin 2pi/2) + 2 = 2
  308. c) f(cos x/3) | F(sin x/3) | F(sin pi/3) = 0 | F(sin pi/3) + 2 = 2
  309.  
  310.  
  311. BERÄKNA LITE INTEGRALER!!!!!!
  312.  
  313. 3201 a) 4§1 2x dx | [x^2]4§1 | 4^2 - 1^2 = 16-1 = 15
  314. b) 2§0 3x^2 dx | [x^3]2§0 | 2^3 - 0^3 = 8 - 0 = 8
  315.  
  316. 3203 a)
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