cos(o)(fcos(x)-fcos(y)+e)=a; sin(o)(fcos(x)-fcos(y)+e)=b; fsin(x)-fsin(y)+l-d=c; tìm x,y

You have a system of equations involving trigonometric functions and parameters. Let's rewrite the system for clarity:

1. \(\cos(\theta)(f \cos(x) - f \cos(y) + e) = a\)
2. \(\sin(\theta)(f \cos(x) - f \cos(y) + e) = b\)
3. \(f \sin(x) - f \sin(y) + l - d = c\)

Here, \(\theta\), \(f\), \(e\), \(l\), and \(d\) are constants, and we need to solve for \(x\) and \(y\).

### Step-by-step process:

1. **From equations 1 and 2**:
   Since both involve the expression \(f \cos(x) - f \cos(y) + e\), we can divide equation 2 by equation 1 to eliminate this expression:
   
   \[
   \frac{\sin(\theta)}{\cos(\theta)} = \frac{b}{a}
   \]
   
   This simplifies to:
   
   \[
   \tan(\theta) = \frac{b}{a}
   \]
   
   Therefore, \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

2. **Using \(\cos(\theta)\) and \(\sin(\theta)\)**:
   Substitute \(\theta\) back into either equation 1 or equation 2 to find the expression \(f \cos(x) - f \cos(y) + e\). Using equation 1:

   \[
   f \cos(x) - f \cos(y) + e = \frac{a}{\cos(\theta)}
   \]

   Since \(\cos(\theta)\) is non-zero (if \(a \neq 0\)), this gives us a simplified expression for \(f \cos(x) - f \cos(y) + e\).

3. **Third equation**:
   We now use the third equation:

   \[
   f \sin(x) - f \sin(y) + l - d = c
   \]

   To solve for \(x\) and \(y\), you can treat the system as a parametric system with respect to \(f\), \(e\), \(l\), \(d\), and constants.

### Final solution outline:
At this point, solving explicitly for \(x\) and \(y\) would require substituting the expressions found and solving the trigonometric equations step-by-step, which depends on the specific values of the parameters.

Would you like more details on solving for \(x\) and \(y\) step-by-step? Let me know!

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Here are five follow-up questions:
1. Would you like me to solve this with specific parameter values?
2. Are there any constraints on \(x\) or \(y\) (e.g., domain restrictions)?
3. Do you need help with a similar system of trigonometric equations?
4. Would you like a geometric interpretation of these equations?
5. Do you want to explore any special cases where some of the constants are zero?

**Tip:** When working with trigonometric equations, leveraging identities like \(\tan(\theta)\) can often simplify complex expressions.

This guide explains how to solve a system of trigonometric equations involving sine and cosine functions with constants. Learn the step-by-step method to find x and y using trigonometric identities and parametric solutions. Suitable for advanced mathematics students.

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