Вариант 1. 1) Вычислите: Sin 10° cos 80° + cos 10°sin 80° CDS 10° COS 35° - Sin 10° sin 35° 2) решите уравнения: a) sin(x-ㄢ)+1=0 2sinx = 53 8) 2sin2x-3sinx-2=0 3) Решите неравенства: a) cos x.>-2 13 2 = LONG H93 BSNWOTENC сопровожд Ceme R 8) sin 2x - 3 принадлежащие 4) Найти все корни уравнения, п Отрезку [0; ЗП] 03 a) sin x = -2

1) Вычислите:

• sin 10° cos 80° + cos 10° sin 80° = sin (10° + 80°) = sin 90° = 1
• cos 10° cos 35° - sin 10° sin 35° = cos (10° + 35°) = cos 45° =$$\frac{\sqrt{2}}{2}$$

2) Решите уравнения:

• a) sin(x - $$\frac{\pi}{3}$$) + 1 = 0

  sin(x - $$\frac{\pi}{3}$$) = -1

  x - $$\frac{\pi}{3}$$ = -$$\frac{\pi}{2}$$ + 2$$\pi$$k, k ∈ Z

  x = -$$\frac{\pi}{2}$$ + $$\frac{\pi}{3}$$ + 2$$\pi$$k, k ∈ Z

  x = -$$\frac{\pi}{6}$$ + 2$$\pi$$k, k ∈ Z

• b) 2sin x = $$\sqrt{3}$$

  sin x = $$\frac{\sqrt{3}}{2}$$

  x = (-1)^k arcsin$$\frac{\sqrt{3}}{2}$$ + $$\pi$$k, k ∈ Z

  x = (-1)^k $$\frac{\pi}{3}$$ + $$\pi$$k, k ∈ Z

• c) 2sin²x - 3sin x - 2 = 0

  Пусть sin x = t, тогда 2t² - 3t - 2 = 0

  D = 9 + 16 = 25

  t₁ = $$\frac{3 + 5}{4}$$ = 2

  t₂ = $$\frac{3 - 5}{4}$$ = -$$\frac{1}{2}$$

  sin x = 2 (не имеет решений, т.к. |sin x| ≤ 1)

  sin x = -$$\frac{1}{2}$$

  x = (-1)^k arcsin(-$$\frac{1}{2}$$) + $$\pi$$k, k ∈ Z

  x = (-1)^k+1$$\frac{\pi}{6}$$ + $$\pi$$k, k ∈ Z

3) Решите неравенства:

• a) cos x > -$$\frac{1}{2}$$

  -$$\frac{2\pi}{3}$$ + 2$$\pi$$k < x < $$\frac{2\pi}{3}$$ + 2$$\pi$$k, k ∈ Z

• b) sin 2x < $$\frac{\sqrt{3}}{2}$$

  -$$\frac{7\pi}{6}$$ + 2$$\pi$$k < 2x < $$\frac{\pi}{3}$$ + 2$$\pi$$k, k ∈ Z

  -$$\frac{7\pi}{12}$$ + $$\pi$$k < x < $$\frac{\pi}{6}$$ + $$\pi$$k, k ∈ Z

4) Найти все корни уравнения, принадлежащие отрезку [0; 3π]

• a) sin x = -$$\frac{\sqrt{3}}{2}$$

  x = (-1)^k+1 $$\frac{\pi}{3}$$ + $$\pi$$k, k ∈ Z

  x₁ = -$$\frac{\pi}{3}$$ (не принадлежит отрезку [0; 3π])

  x₂ = $$\pi$$ + $$\frac{\pi}{3}$$ = $$\frac{4\pi}{3}$$

  x₃ = 2$$\pi$$ - $$\frac{\pi}{3}$$ = $$\frac{5\pi}{3}$$

  x₄ = 3$$\pi$$ + $$\frac{\pi}{3}$$ = $$\frac{10\pi}{3}$$

Ответ:

• 1) 1, $$\frac{\sqrt{2}}{2}$$
• 2) a) x = -$$\frac{\pi}{6}$$ + 2$$\pi$$k, k ∈ Z; b) x = (-1)^k $$\frac{\pi}{3}$$ + $$\pi$$k, k ∈ Z; c) x = (-1)^k+1$$\frac{\pi}{6}$$ + $$\pi$$k, k ∈ Z
• 3) a) -$$\frac{2\pi}{3}$$ + 2$$\pi$$k < x < $$\frac{2\pi}{3}$$ + 2$$\pi$$k, k ∈ Z; b) -$$\frac{7\pi}{12}$$ + $$\pi$$k < x < $$\frac{\pi}{6}$$ + $$\pi$$k, k ∈ Z
• 4) $$\frac{4\pi}{3}$$, $$\frac{5\pi}{3}$$, $$\frac{10\pi}{3}$$