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resuelve el sistema para (φ,θ,ψ)(φ,θ,ψ)

The system of equations you uploaded is:

1. \(\cos \phi \cos \theta = -\sin(q^2 + q^3) \cdot \cos(q^4)\)
2. \(\sin \phi \cos \theta = \cos(q^2 + q^3) \cdot \cos(q^4)\)
3. \(-\sin \theta = \sin(q^4)\)
4. \(\cos \theta \sin \psi = \cos(q^4)\)
5. \(\cos \theta = 0\)

We need to solve this system for \(\phi\), \(\theta\), and \(\psi\). Let's approach this step by step:

### Step 1: Start with the fifth equation
From equation 5:
\[
\cos \theta = 0
\]
This implies that:
\[
\theta = \frac{\pi}{2} + n\pi \quad \text{for} \, n \in \mathbb{Z}
\]
This simplifies \(\cos \theta\) and \(\sin \theta\):
- \(\cos \theta = 0\)
- \(\sin \theta = (-1)^n\)

### Step 2: Analyze the third equation
Using the third equation:
\[
-\sin \theta = \sin(q^4)
\]
Since \(\sin \theta = (-1)^n\), we have:
\[
-(-1)^n = \sin(q^4)
\]
Thus:
\[
(-1)^{n+1} = \sin(q^4)
\]
This means that \(n\) is chosen such that \((-1)^{n+1} = \sin(q^4)\). Therefore, \(n = 1\) if \(\sin(q^4) = 1\), and so on.

### Step 3: Use the first and second equations
Since \(\cos \theta = 0\), the left-hand sides of equations 1 and 2 both equal 0:
1. \(\cos \phi \cdot 0 = -\sin(q^2 + q^3) \cdot \cos(q^4)\)
2. \(\sin \phi \cdot 0 = \cos(q^2 + q^3) \cdot \cos(q^4)\)

Both these equations hold true regardless of the values of \(\phi\), meaning they don't provide further information to determine \(\phi\).

### Step 4: Analyze the fourth equation
From the fourth equation:
\[
\cos \theta \sin \psi = \cos(q^4)
\]
Since \(\cos \theta = 0\), this equation simplifies to:
\[
0 \cdot \sin \psi = \cos(q^4)
\]
This implies \(\cos(q^4) = 0\).

### Step 5: Final conditions
From \(\cos(q^4) = 0\), we know:
\[
q^4 = \frac{\pi}{2} + m\pi \quad \text{for} \, m \in \mathbb{Z}
\]
Thus, we have the following:
- \(\theta = \frac{\pi}{2} + n\pi\) (where \(n\) satisfies \((-1)^{n+1} = \sin(q^4)\))
- \(\phi\) remains undetermined from these equations.
- \(\psi\) can be any value, but \(\sin \psi = 0\) is a possibility if \(\cos(q^4) = 0\).

---

Would you like more details on any of the steps, or do you have any questions? 

Here are 5 relative questions for practice:
1. What is the general solution for \(\cos \theta = 0\)?
2. How do we solve trigonometric equations involving \(\sin \theta = 1\)?
3. What values of \(q^4\) satisfy \(\cos(q^4) = 0\)?
4. How does the periodicity of trigonometric functions affect the solutions?
5. Can we use different \(n\) or \(m\) values to determine specific solutions?

**Tip:** Always check for special trigonometric identities or values, such as \(\cos \theta = 0\), to simplify complex systems of equations.

Explore how to solve a complex system of trigonometric equations involving cos, sin, and periodic functions. This solution covers trigonometry, periodic functions, and trigonometric identities, suitable for advanced high school or university students.

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