23.03.26 ① Sinx = 2 (2) cosx = 1 メニー ⑥ 2003x+√3=0 ⑥ 25mx-5 ④ 25m²x-sinx-3:0 (125mx13 ⑤ 6 cosx+toosx-3=0 (5) 2005x+50 ⑥) 35m²x + sinxcosx-2003 (6) 25m²x +3

I столбик

1. sin(x) = 1/2

   x = (-1)^n * arcsin(1/2) + \( \pi \)n, n \( \in \) Z

   x = (-1)^n * \( \pi \)/6 + \( \pi \)n, n \( \in \) Z

2. cos(x) = 1

   x = 2\( \pi \)n, n \( \in \) Z

3. 2cos(x) + \( \sqrt{3} \) = 0

   cos(x) = -\( \sqrt{3} \)/2

   x = \( \pm \) arccos(- \( \sqrt{3} \)/2) + 2\( \pi \)n, n \( \in \) Z

   x = \( \pm \) 5\( \pi \)/6 + 2\( \pi \)n, n \( \in \) Z

4. 2sin²(x) - sin(x) - 3 = 0

   Пусть t = sin(x), тогда:

   2t² - t - 3 = 0

   D = (-1)² - 4 * 2 * (-3) = 1 + 24 = 25

   t₁ = (1 + \( \sqrt{25} \))/(2*2) = (1 + 5)/4 = 6/4 = 3/2 > 1 (не подходит, так как |sin(x)| ≤ 1)

   t₂ = (1 - \( \sqrt{25} \))/(2*2) = (1 - 5)/4 = -4/4 = -1

   sin(x) = -1

   x = -\( \pi \)/2 + 2\( \pi \)n, n \( \in \) Z

5. 6cos²(x) + cos(x) - 3 = 0

   Пусть t = cos(x), тогда:

   6t² + t - 3 = 0

   D = 1² - 4 * 6 * (-3) = 1 + 72 = 73

   t₁ = (-1 + \( \sqrt{73} \))/12

   t₂ = (-1 - \( \sqrt{73} \))/12

   cos(x) = (-1 + \( \sqrt{73} \))/12

   x = \( \pm \) arccos((-1 + \( \sqrt{73} \))/12) + 2\( \pi \)n, n \( \in \) Z

   cos(x) = (-1 - \( \sqrt{73} \))/12

   x = \( \pm \) arccos((-1 - \( \sqrt{73} \))/12) + 2\( \pi \)n, n \( \in \) Z

6. 3sin²(x) + sin(x)cos(x) - 2cos²(x) = 0

   Разделим обе части уравнения на cos²(x) (при условии, что cos(x) ≠ 0):

   3tan²(x) + tan(x) - 2 = 0

   Пусть t = tan(x), тогда:

   3t² + t - 2 = 0

   D = 1² - 4 * 3 * (-2) = 1 + 24 = 25

   t₁ = (-1 + \( \sqrt{25} \))/(2*3) = (-1 + 5)/6 = 4/6 = 2/3

   t₂ = (-1 - \( \sqrt{25} \))/(2*3) = (-1 - 5)/6 = -6/6 = -1

   tan(x) = 2/3

   x = arctan(2/3) + \( \pi \)n, n \( \in \) Z

   tan(x) = -1

   x = -\( \pi \)/4 + \( \pi \)n, n \( \in \) Z

II столбик

1. cos(x) = -1/2

   x = \( \pm \) arccos(-1/2) + 2\( \pi \)n, n \( \in \) Z

   x = \( \pm \) 2\( \pi \)/3 + 2\( \pi \)n, n \( \in \) Z

2. sin(x) = -\( \sqrt{2} \)/2

   x = (-1)^n * arcsin(- \( \sqrt{2} \)/2) + \( \pi \)n, n \( \in \) Z

   x = (-1)^n * (- \( \pi \)/4) + \( \pi \)n, n \( \in \) Z

   x = (-1)^(n+1) * \( \pi \)/4 + \( \pi \)n, n \( \in \) Z

3. 2sin(x) - \( \sqrt{3} \) = 0

   sin(x) = \( \sqrt{3} \)/2

   x = (-1)^n * arcsin(\( \sqrt{3} \)/2) + \( \pi \)n, n \( \in \) Z

   x = (-1)^n * \( \pi \)/3 + \( \pi \)n, n \( \in \) Z

4. 2sin(x) + 3 = 0

   sin(x) = -3/2

   Решений нет, так как |sin(x)| ≤ 1

5. 2cos²(x) + 5cos(x) = 0

   cos(x) * (2cos(x) + 5) = 0

   cos(x) = 0

   x = \( \pi \)/2 + \( \pi \)n, n \( \in \) Z

   2cos(x) + 5 = 0

   cos(x) = -5/2

   Решений нет, так как |cos(x)| ≤ 1

6. 2sin²(x) + 3 = 0

   2sin²(x) = -3

   sin²(x) = -3/2

   Решений нет, так как sin²(x) ≥ 0