Алгебра и нач анализа_Реш экз зад 11кл из Сборн заданий для экз_Дорофеев_Решения (991497), страница 21
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sin2x + sin6x = 3cos2x;2sin4x cos2x – 3cos2x = 0; cos2x(2sin4x – 3) = 0;cos2x = 0или2sin4x – 3 = 0;π3sin 4 x = - решений нет2 x = + π m, m ∈ Z ;22x=π4+π2т.к. |sin α| ≤ 1;m, m ∈ Z .π(1 + 2m ) , где m ∈ Z.46.41. 144cos4x – 4sin4x = 9sin22x;4sin4 x + 36sin2x ⋅ cos2x + 81cos4x – 225cos4x = 0;(2sin2x + 9cos2x – 15cos2x)(2sin2x + 9cos2x + 15cos2x) = 0;sin2x – 3cos2x = 0 ⏐:cos2x или 11cos2x + 1 = 0Ответ:tgx = 0 ± 3πилирешений нет;πОтвет: ± π n, n ∈ Z .336.42. 2(cos4x – sin x ⋅ cos3x) = sin4x + sin2x;2(cos4x – sin x ⋅ cos3x) = 2sin3x cos x;cos4x = sin3x cos x + sin x cos3x; cos4x = sin4x | : cos4x ≠ 0;x=±+ π n, n ∈ Z ;πππππ+ π n; x = + n, n ∈ Z .
Ответ:+ n, n ∈ Z .416 416 46.43. cos7x + cos x = 2cos3x(sin2x – 1);2cos4x cos3x – 2cos3x(sin2x – 1) = 0; cos3x(cos4x + 1 – sin2x) = 0;cos3x = 0или2cos22x – sin2x = 0;tg4x = 1, 4 x =3x =x=π2π6+ π m, m ∈ Z ;+π32sin22x + sin2x – 2 = 0;π m, m ∈ Z ;213⎡−1 − 17< −1 – решений нет, т.к. |sin α| ≤ 1,⎢sin 2 x =4⎢−1 + 17117 − 1 π⎢k+ k, k ∈ Z.; x = ( −1) arcsin⎢⎣sin 2 x =4242117 − 1 πk+ k , m, k ∈ Z.(1 + 2m ) ; ( −1) arcsin62426.44. cos5x – cos x = sin3x(2cos4x + 1);1⎞⎛sin 3x sin 2 x + sin 3x ⎜ cos 4 x + ⎟ = 0;2⎠⎝Ответ:π1⎞⎛sin 3x ⎜ sin 2 x + 1 − 2sin 2 2 x + ⎟ = 0;2⎠⎝sin3x = 0; 3x = πm, m ∈ Z; x =πm, m ∈ Z ;3или 2sin22x – sin2x – 1,5 = 0;⎡1 − 1311 − 13 πk< 1; x = ( −1) arcsin+ k, k ∈ Z ,⎢sin 2 x =4242⎢1 + 13⎢> 1 – решений нет.⎢⎣sin 2 x =4Ответ:π3m;113 − 1 πk +1+ k , m, k ∈ Z.( −1) arcsin2426.45.
cos3 x − sin x = 3 ( cos x − sin 3x ) ;cos3 x + 3 sin 3x = sin x + 3 cos x;⎛1⎞⎛1⎞333 + 1 ⎜⎜ cos3 x +sin 3 x ⎟⎟ = 3 + 1 ⎜⎜ sin x +cos x ⎟⎟ ;2222⎝⎠⎝⎠⎛π⎞⎛π⎞sin ⎜ + 3x ⎟ = sin ⎜ + x ⎟ ;⎝6⎠⎝3⎠π2sin 6+ 3x −2π3−xπ⋅ cos 6+ 3x +π ⎞ ⎛π⎛⎞sin ⎜ x − ⎟ cos ⎜ + 2 x ⎟ = 0 ;12 ⎠⎝⎝4⎠2142π3+x= 0 | : 2;⎛π⎞cos ⎜ + 2 x ⎟ = 0⎝4⎠π4+ 2x =2x =x=ππ84π2π ⎞⎛sin ⎜ x − ⎟ = 0;12 ⎠⎝или+ π m, m ∈ Z ;x−+ π m, m ∈ Z ;π+2m, m ∈ Z ;x=Ответ:π8π12π12(1 + 4m ) ;= π k, k ∈ Z;+ π k, k ∈ Z.π12(1 + 12k ) , k, m ∈ Z.6.46.
cos 2 x = 2 ( cos x − sin x ) ;(cos x − sin x)(cos x + sin x − 2) = 0;cos x – sin x = 0cos x + sin x − 2 = 0;или11cos x +sin x = 1;22tg x = 1;x=π4+ π m, m ∈ Z ;ππ⎛π⎞sin ⎜ + x ⎟ = 1; + x = + 2π k , k ∈ Z ;42⎝4⎠x=π4+ 2π k , k ∈ Z .π(1 + 4m ) , m ∈ Z.46.47. sin x ⋅ cos3x = sin2x;sin x cos3x = 2sin x cos x; sin x(cos3x – 2cos x) = 0;sin x = 0илиcos3x – 2cos x = 0;x = πm, m ∈ Z;cos x(4cos2x – 5) = 0;cos x = 0или4cos2x = 5;π5x = + π k, k ∈ Z;cos 2 x = > 1;24решений нет.Ответ:Ответ: πm;π+ π k , m, k ∈ Z.26.48. 5sin4x – cos4x = sin22x;22⎛ 1 − cos 2 x ⎞ ⎛ 1 + cos 2 x ⎞25⎜⎟ −⎜⎟ = sin 2 x;22⎝⎠ ⎝⎠5 – 10cos2x + 5cos22x – 1 – 2cos2x – cos22x = 4sin22x;2154cos22x – 12cos2x + 4 = 4 – 4cos22x | : 4;2cos22x – 3cos2x = 0; cos2x(2cos2x – 3) = 0;cos2x = 0; 2 x =π2+ π k, k ∈ Z; x =или 2cos2x – 3 = 0; cos 2 x =π4+π2k, k ∈ Z;3> 1 - решений нет, т.к.
|cos α| ≤ 1.2π(1 + 2k ) , k ∈ Z .46.49. sin 6x + sin24x = 1; 1–cos12x+1–cos8x=2; cos12x + cos8x = 0;2cos10x cos2x = 0;Ответ:2⎡cos10 x = 0,⎢⎣cos 2 x = 0;Ответ:π20π⎡⎢ x = 20 (1 + 2m ) , m ∈ Z ,⎢⎢ x = π (1 + 2k ) , k ∈ Z .4⎣⎢π⎡⎢10 x = 2 + π m,⎢⎢2x = π + π k ,⎢⎣2(1 + 2l ) , l ∈ Z .1; sin22x = 1;sin x cos x22sin 2x = 2; 1 – cos4x = 2; cos4x = -1; 4x = π + 2πn;6.50. 2sin2x = tg x + ctg x; 2sin 2 x =ππ(1 + 2n ) , n ∈ Z . Ответ: (1 + 2n ) , n ∈ Z .446.51. sin5x = sin x + sin2x;2cos3x sin2x – sin2x = 0; sin2x(2cos3x – 1) = 0;x=sin2x = 0; 2x = πm, m ∈ Z; x =π2m, m ∈ Z ;1π 2или 2cos3x – 1 = 0; cos3x = ; x = ± + π n, n ∈ Z .29 3ππ 2Ответ: m; ± + π n, m, n ∈ Z.29 36.52. 6sin2x + 2sin22x = 5; 3(1 – cos2x) + 2(1 – cos22x) = 5;3 – 3cos2x + 2 – 2cos22x = 5; 2cos22x + 3cos2x = 0;cos2x = 0или2cos2x = -3;π32 x = + π m, m ∈ Z ;cos 2 x = − < −1 – решений нет;22ππx = (1 + 2m ) , m ∈ Z .
Ответ: (1 + 2m ) , m ∈ Z .442166.53. cos26x – sin23x – 1 = 0;1 − cos 6 xcos 2 6 x −− 1 = 0; 2cos26x + cos6x – 3 = 0.23Пусть cos6x = y, тогда 2у2 + у – 3 = 0; у1 = 1, y2 = − ;2πcos6x = 1; 6x = 2πn, n ∈ Z; x = n, n ∈ Z ;33или cos 6 x = − < −1 - решений нет, т.к. |cos α| ≤ 1.2πОтвет: n, n ∈ Z .36.54. cos x – cos3x = 3sin2x;2sin2x sin x = 3sin2x; 4sin2x cos x – 3sin2x = 0; sin2x(4cos x – 3) = 0;или4cos x – 3 = 0;sin2x = 033x = πm, m ∈ Z;cos x = ; x = ± arccos + 2π k , k ∈ Z .443Ответ: πm, ± arccos + 2π k , m, k ∈ Z.425426.55. cos 2 x + 6cos 2 x = ;1616cos42x + 96cos22x – 25 = 0.
Пусть cos22x = y, тогда116у2 + 96у – 25 = 0; D = 482 + 25 ⋅ 16 = 2304 + 400 = 2704;4−48 − 5225−48 + 52 1y1 == − , y2 == ;16416425112илиcos 2 x = −cos 2 2 x = ; 2cos 2 2 x = ;442111 + cos 4 x = ; cos 4 x = − ;решений нет;222π π4 x = ± π + 2π k ; x = ± + k , k ∈ Z .36 2π πОтвет: ± + k , k ∈ Z .6 21 − cos 2 x1 + cos 2 x2−8+ 1 = 0;6.56. 3tg x – 8cos2x + 1 = 0; 31 + cos 2 x2217π+ π k, k ∈ Z;23 – 3cos2x – 4 – 8cos2x – 4cos22x + 1 + cos2x = 0;4cos22x + 10cos2x = 0 | : 4; cos2x(cos2x + 2,5) = 0;cos2x = 0илиcos2x + 2,5 = 0;cos2x ≠ -1, x ≠2x =x=ππ42+ π m, m ∈ Z ;+Ответ:π2π4cos2x = -2,5 – нет решений,|cos α| ≤ 1.m, m ∈ Z ;(1 + 2m ) , m ∈ Z.sin 2 x+ 4cos 2 x = 7; cos2x ≠ 0;cos 2 x24242sin x + 4cos x – 7cos x = 0; 4cos x – 7cos2x – 2cos2x + 2 = 0;4cos4x – 9cos2x + 2 = 0; cos2x = t; 4t2 – 9t + 2 = 0;9−7 19+7= , t2 == 2;D = 81 – 32 = 49; t1 =8481cos 2 x =илиcos2x = 2 – решений нет,411т.к.
|cos α| ≤ 1;2cos 2 x = ; 1 + cos 2 x = ;221ππОтвет: ± + π n, n ∈ Z .cos 2 x = − ; x = ± + π n, n ∈ Z .3236.58. ctg2x – 8sin2x = 1; sin x ≠ 0; x ≠ πn, n ∈ Z;cos 2 x− 8sin 2 x = 1; cos2x – 8sin4x – sin2x = 0;sin 2 x8sin4x + sin2x – 1 + sin2x = 0; 8sin4x + 2sin2x – 1 = 0;11илиsin 2 x =sin 2 x = − - решений нет;4211π1 − cos 2 x = ;cos 2 x = ; x = ± + π n, n ∈ Z .2266.57. 2tg2x + 4cos2x = 7; 2π+ π n, n ∈ Z .66.59. 9ctg2x + 4sin2x = 6; 9cos2x + 4sin4x – 6sin2x = 0;4sin4x + 9 – 9sin2x – 6sin2x = 0; 4sin4x – 15sin2x + 9 = 0;Ответ: ±218Пусть sin2x = y, тогда 4у2 – 15у + 9 = 0; D = 225 – 144 = 81;15 − 9 315 + 9y1 == , y2 == 3;8483sin 2 x =или sin2x = 3 – решений нет, т.к.
|sin α| ≤ 1;43ππ; x = ± + π n, n ∈ Z . Ответ: x = ± + π n, n ∈ Z .2336.60. 1 – cos6x = tg3x; 2sin23x = tg3x; sin3x(2sin3x cos3x – 1) = 0;sin3x(sin6x – 1) = 0;πsin3x = 0; 3x = πm, m ∈ Z; x = m, m ∈ Z ;3ππ πили sin6x–1 = 0; sin6x = 1; 6 x = + 2π n, n ∈ Z ; x = + n, n ∈ Z .212 3ππ π+ n, m, n ∈ Z.Ответ: m,312 36.61.
cos x – cos3x = sin2x;2sin2x ⋅ sin x – sin2x = 0; sin2x(2sin x – 1) = 0;πsin2x = 0; 2x = πm, m ∈ Z; x = m, m ∈ Z ;21k π+ π k, k ∈ Z.или 2sin x – 1 = 0; sin x = ; x = ( −1)62πk π+ π k , m, k ∈ Z.Ответ: m; ( −1)266.62. cos2x – cos4x = sin6x;2sin3x sin x – 2sin3x cos3x = 0; sin3x(sin x – cos3x) = 0;πsin3x = 0; 3x = πm, m ∈ Z; x = m, m ∈ Z ;3sin x = ±π⎞⎛π⎞⎛π⎞ ⎛или sinx–cos3x=0; sin x − sin ⎜ − 3x ⎟ = 0; 2cos ⎜ − x ⎟sin ⎜ 2x − ⎟ = 0;4⎠⎝2⎠⎝4 ⎠ ⎝⎡⎛π⎞⎢ cos ⎜ 4 − x ⎟ = 0,⎝⎠⎢π ⎞⎛⎢⎢ sin ⎜⎝ 2 x − 4 ⎟⎠ = 0;⎣Ответ:π3m; π k −π⎡π⎢ 4 − x = 2 + π k,⎢⎢ 2 x − π = π n;4⎣⎢π4;π8π⎡⎢x = − 4 + π k,k ∈ Z ,⎢⎢ x = π + π n, n ∈ Z .⎢⎣82(1 + 4n ) , m, k, n ∈ Z.219xx− sin 4 ;22xxsin 2 x = cos 2 − sin 2 ; sin2x = cos x; cos x(2sin x – 1) = 0;226.63.
sin 2 x = cos 4⎡cos x = 0,⎢⎣ 2sin x = 1;ππ⎡⎢ x = 2 + π m, m ∈ Z ,⎢⎢ x = ( −1)k π + π k , k ∈ Z .⎢⎣6(1 + 2m ) ; ( −1)kπ+ π k , m, k ∈ Z.6xx6.64. sin 2 x = cos 4 − sin 2 ; sin2x = cos x; 1 – cos2x = cos x;22cos2x + cos x – 1 = 0;Ответ:cos x =2−1 + 52x = ± arccoscos x =или5 −1+ 2π k , k ∈ Z ;2−1 − 5< −1;2нет решений, т.к. |cosα|≤1.5 −1+ 2π k , k ∈ Z ;26.65.
cos2x = 2(cos x – sin x); cos2x – sin2x = 2(cos x – sin x);(cos x – sin x)(cos x + sin x – 2) = 0;cos x – sin x = 0илиcos x + sin x = 2;Ответ: ± arccos⎛π⎞2 sin ⎜ − x ⎟ = 0;⎝4⎠π4− x = π k;x=π4⎛π⎞2 sin ⎜ + x ⎟ = 2;⎝4⎠⎛π⎞sin ⎜ + x ⎟ = 2 - решений нет,⎝4⎠− π k, k ∈ Z;Ответ:π4т.к. |cos α| ≤ 1.− π k, k ∈ Z ;6.66. (cos6x – 1)ctg3x = sin3x; sin3x ≠ 0, x ≠−2sin 2 3x220cos3x− sin 3x = 0; ⏐sin3xsin 3 xπ3k, k ∈ Z;2cos3x = −1;3 x = ±2π+ 2π n, n ∈ Z322x = ± π + π n, n ∈ Z9322Ответ: ± π + π n, n ∈ Z .936.67.
sin x sin5x = cos4x; cos4x – cos6x = 2cos4x;cos4x + cos6x = 0; 2cos5x cos x = 0;π⎡⎢5 x = 2 + π m,⎢⎢ x = π + π k;⎣⎢2⎡cos5 x = 0,⎣⎢cos x = 0;πππ π⎡⎢ x = 10 + 5 m, m ∈ Z ,⎢⎢x = π + π k, k ∈ Z.⎣⎢2π+ m;+ π k , m, k ∈ Z.10 526.68. cos x cos3x = cos2x; cos4x + cos2x – 2cos2x = 0;cos4x – cos2x = 0; -2sin3x sin x = 0;sin3x = 0илиsin x = 0;x = πn, n ∈ Z.3x = πm, m ∈ Z;Ответ:x=ππm, m ∈ Z ;Ответ: k , k ∈ Z .336.69.
3cos x + 2tg x = 0; cos x ≠ 0; 3 cos2x + 2sin x = 0;3 – 3sin2x + 2sin x = 0; 3sin2x – 2sin x – 3 = 0.DПусть sin x = y, тогда имеем: 3у2 – 2у – 3 = 0;= 1 + 9 = 10;4y1 =1 − 10;3sin x =y2 =1 − 103x = ( −1)k +11 + 10;3илиarcsinsin x =1 + 10, решений нет,310 − 1+ π k , k ∈ Z ; т.к. |sin α| ≤ 1.310 − 1+ π k, k ∈ Z;36.70. 5sin x – 4ctg x = 0, sin x ≠ 0;5 – 5cos2x – 4cos x = 0; 5cos2x + 4cos x – 5 = 0;cos x = y;Ответ: ( −1)k +1arcsin2215у2 + 4у – 5 = 0;y1 =D= 4 + 25 = 29;4−2 − 29;529 − 25cos x =x = ± arccosy2 =или−2 + 29;5cos x =−2 − 29< −1, решений нет,529 − 2+ 2π k , k ∈ Z ;5т.к.
|sin α| ≤ 1.29 − 2+ 2π k , k ∈ Z .5226.71. 8sin x + 4sin 2x = 5 – 8cos2x; 4(1 – cos2x) + 4(1 – cos22x) ++ 8cos2x – 5 = 0; 4cos22x – 4cos2x – 3 = 0;13cos 2 x = > 1 - нет решений,илиcos 2 x = −22Ответ: ± arccosx=±π3+ π m, m ∈ Z ;т.к. |cos α| ≤ 1.π+ π m, m ∈ Z .36.72. 2sin x = 4sin22x + 7cos2x – 6;1 – cos2x – 4 + 4cos22x – 7cos2x + 6 = 0;4cos22x – 8cos2x + 3 = 0, пусть cos2x = y, тогда13D4у2 – 8у + 3 = 0;= 16 − 12 = 4; y1 = , y2 = ;42211илиcos 2 x =cos 2 x = 1 - решений нет,22Ответ: ±22x = ±x=±ππ63+ 2π m, m ∈ Z ;+ π m, m ∈ Z .т.к.
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