find all the solutions of 2sin(2x)+cos(2x)=-2

To find all the solutions to the equation \( 2\sin(2x) + \cos(2x) = -2 \), we can follow these steps:

### Step 1: Consider the range of the trigonometric functions
The values of \( \sin(2x) \) and \( \cos(2x) \) range between \(-1\) and \(1\).

### Step 2: Analyze the equation
Given \( 2\sin(2x) + \cos(2x) = -2 \), let's consider the extreme values these functions can achieve:

- The minimum value of \( 2\sin(2x) \) is \( -2 \times 1 = -2 \).
- The minimum value of \( \cos(2x) \) is \( -1 \).

To satisfy \( 2\sin(2x) + \cos(2x) = -2 \), the expression must equal \( -2 \). This means:

\[
2\sin(2x) = -2 \quad \text{and} \quad \cos(2x) = 0
\]

### Step 3: Solve \( 2\sin(2x) = -2 \)
\[
\sin(2x) = -1
\]

The general solutions for \( \sin(2x) = -1 \) are:
\[
2x = \frac{3\pi}{2} + 2k\pi \quad \text{for integer } k
\]
\[
x = \frac{3\pi}{4} + k\pi
\]

### Step 4: Solve \( \cos(2x) = 0 \)
The general solutions for \( \cos(2x) = 0 \) are:
\[
2x = \frac{\pi}{2} + n\pi \quad \text{for integer } n
\]
\[
x = \frac{\pi}{4} + \frac{n\pi}{2}
\]

### Step 5: Find the intersection of solutions
For \( \sin(2x) = -1 \) and \( \cos(2x) = 0 \) to be satisfied simultaneously:

Let's check for each solution from \( \sin(2x) = -1 \):

- For \( x = \frac{3\pi}{4} + k\pi \), where \( k = 0 \), \( \cos(2x) = \cos\left(\frac{3\pi}{2}\right) = 0 \).

Thus, the solution that satisfies both equations is:

\[
x = \frac{3\pi}{4} + k\pi \quad \text{for integer } k
\]

### Conclusion
The general solution for the equation \( 2\sin(2x) + \cos(2x) = -2 \) is:
\[
x = \frac{3\pi}{4} + k\pi \quad \text{for integer } k
\]

This represents all the solutions to the given equation.

---

Would you like more details or have any questions?

### Further Questions:
1. What is the general solution for \( \sin(2x) = 1 \)?
2. How do the solutions differ for \( 2\sin(2x) + \cos(2x) = 2 \)?
3. Can you solve \( 2\sin(x) + \cos(x) = -2 \) for all solutions?
4. What is the effect of adding a constant to the equation on the solutions?
5. How would the solution change if the equation were \( 2\sin(2x) - \cos(2x) = -2 \)?

### Tip:
Always check the domain and range of trigonometric functions when solving such equations to ensure the solutions are valid.

Explore the detailed solution to the trigonometric equation 2sin(2x) + cos(2x) = -2. Learn about double angle identities and how they are applied to find all solutions. Suitable for advanced high school students and above.

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