КАРТОЧКА 4 1. cos t = a ; |a| ≤ 1; t = ± arccos a + 2πn. 2. а) 1 + cos x = 2sin2 x ; cos 2x + cos x = 0; 3x x π 2 πn cos = 0; x= + ; x = π + 2πn; cos 2 2 3 3 π π б) sin 2x + 2 3 cos2 x = 0; cos x(sin x + 3 cos x) = 0; x = + πk; x = – + πk. 2 3
π⎞ 2 ⎛ 3. sin ⎜ x + ⎟ ≤ ; ⎝ 4⎠ 2
⎡ 3π ⎤ + 2 πn; 2 πn ⎥ . x ∈ ⎢− ⎣ 2 ⎦ π ⎧ π πn π ⎧ y= + ⎪⎪ x = 2 − y ⎪x + y = 4 2 . 2 4. ⎨ ; ⎨ ; π πn 2 ⎪sin2 x + cos2 y = 1 ⎪cos y = ± x= − ⎩ ⎪⎩ 4 2 2 π π 5. 2x − π (sin x – 1) = 0; ОДЗ: x ≥ ; sin x = 1; x = + 2πn; n = 0; 1; 2; 3. 2 2
КАРТОЧКА 5. 1. sin t = a; |a| ≤ 1; t = (–1)k arcsin a + πk. 2. а) 1 – cos 2x + sin x = 0; 2sin2 x + sin x = 0; 1 π sin x = 0; x = πn; sin x = – ; x = (–1)k+1 + πk; 2 6 x x x 2 x б) 5sin 2x – 6cos x = 6; 12cos – 10sin cos = 0 cos = 0; 2 2 2 2 x 6 x 6 x = 2arctg + 2πn. x = 2πn + π; sin = cos ; 2 5 2 5 ⎡ 2π ⎤ 3. tg 2x ≥ – 3 ; x ∈ ⎢− + 2 πn; π + 2 πn⎥ . 3 ⎣ ⎦ π ⎧ ⎧ x = π − y ⎪ y = ( −1) k + πk ⎧x + y = π ⎪ ⎪ 6 ; ⎨ 4. ⎨ 1 ; ⎨ ⎩sin x + sin y = −1 ⎪ x = π − ( −1) k π − πk ⎪⎩sin y = 2 ⎪⎩ 6 π 5. |x|sin x + x = 0, т.к. x > 0, то sin x = –1; x = – + 2πn; n ∈ N. 2
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