Question

√3sinx+2sin(2x+П/6)=√3sin2x-1 ...[-3П;-3П/2]

Answer 1

Ответ:   - 5π/3,   - 2π,   -3π.

Пошаговое объяснение:

√3sinx + 2sin(2x + π/6) = √3sin2x + 1

√3sinx + 2sin2x · cos(π/6) + 2cos2x · sin(π/6) = √3sin2x + 1

√3sinx + 2sin2x · √3/2 + 2 · cos2x · 1/2 = √3sin2x + 1

√3sinx + √3sin2x + cos2x = √3sin2x + 1

√3sinx  + cos2x =  1

√3sinx + 1 - 2sin²x = 1

2sin²x - √3sinx = 0

sinx (2sinx - √3) = 0

1) sinx = 0

  x = πn,   n∈Z

2) sinx = √3/2

   x = π/3 + 2πk,   k∈Z                x = 2π/3 + 2πm,     m∈Z

x ∈ [- 3π; - 3π/2]:

1)

-3π ≤ πn ≤ -3π/2

-3 ≤ n ≤ -1,5

n ∈ Z, ⇒  n = - 3     x = - 3π

               n = - 2     x = - 2π

2)

- 3π ≤ π/3 + 2πk ≤ - 3π/2

- 10π/3 ≤ 2πk ≤ - 11π/6

- 5/3 ≤ k ≤ - 11/12

k ∈ Z,   ⇒   k = - 1     x = - 5π/3

- 3π ≤ 2π/3 + 2πm ≤ - 3π/2

- 11π/3 ≤ 2πm ≤ - 13π/6

- 11/6 ≤ m ≤ - 13/12

m ∈ Z, нет целых значений m на промежутке.

Answer 2

To solve the equation √3sinx + 2sin(2x+π/6) = √3sin2x - 1 over the interval [-3π, -3π/2], we'll first simplify the equation and then find the solutions within the given interval.

Let's simplify the equation step-by-step:

Step 1: Apply the double angle identity for sine: sin(2x) = 2sin(x)cos(x).

√3sinx + 2sin(2x+π/6) = √3sin2x - 1

Step 2: Replace sin(2x) with 2sin(x)cos(x).

√3sinx + 2(2sin(x)cos(x) + π/6) = √3sin2x - 1

Step 3: Distribute 2 to the terms inside the parenthesis.

√3sinx + 4sin(x)cos(x) + π/3 = √3sin2x - 1

Step 4: Move all terms to the left side of the equation.

√3sinx + 4sin(x)cos(x) - √3sin2x - 1 + π/3 = 0

Step 5: Apply the double angle identity for sine: sin(2x) = 2sin(x)cos(x).

√3sinx + 4sin(x)cos(x) - √3 * 2sin(x)cos(x) - 1 + π/3 = 0

Step 6: Combine like terms.

√3sinx + 4sin(x)cos(x) - 2√3sin(x)cos(x) - 1 + π/3 = 0

Step 7: Factor out sin(x).

sin(x) (√3 + 4cos(x) - 2√3cos(x)) - 1 + π/3 = 0

Step 8: Factor out cos(x) and substitute π with 180 degrees.

sin(x) (√3 + (4 - 2√3)cos(x)) - 1 + 60° = 0

Now, we can see that this equation is a bit complex to solve analytically. Therefore, we'll use numerical methods or a graphing tool to find approximate solutions. Let's use a graphing tool to visualize the equation and find the solutions within the given interval.

Plotting the equation y = √3sinx + 2sin(2x+π/6) - (√3sin2x - 1) over the interval [-3π, -3π/2], we get the following graph:

(Note: As an AI language model, I can't provide real-time graphs, but you can use graphing tools like Desmos or other graphing calculators to plot the equation.)

By analyzing the graph, you can approximate the x-values where the function crosses the x-axis (i.e., where y = 0). These x-values are the solutions to the equation over the given interval.

Keep in mind that the exact solutions might be difficult to obtain without using numerical methods due to the complexity of the equation. Numerical methods like the Newton-Raphson method or the bisection method can provide more accurate solutions if needed.