https://file.aiquickdraw.com/mathbot/file/1725285548181sgtp6ake.jpg

Cos theta+8sin theta= 4 is the original question,may u review on it

Let's review the image and solve the problem step by step.

### Given Equations
From the image, the given trigonometric system is:

\[
\begin{cases}
\cos\theta + 8\sin\theta = 4 & \text{(Equation 1)} \\
x\cos\theta - \sin\theta = 1 & \text{(Equation 2)}
\end{cases}
\]

### The System of Equations:

We have two equations that need to be solved together. Here's how to approach them:

1. **First Equation Analysis**: 
   \[
   \cos\theta + 8\sin\theta = 4
   \]

   This equation can be rewritten using the identity:
   \[
   R\cos(\theta - \alpha) = R\left(\frac{4}{R}\right)
   \]
   Where \( R = \sqrt{1^2 + 8^2} = \sqrt{65} \) and \(\alpha\) is the angle whose tangent is \(\tan\alpha = \frac{8}{1} \).

2. **Second Equation Analysis**:
   \[
   x\cos\theta - \sin\theta = 1
   \]
   
   This equation can be similarly approached by isolating \(x\) or using trigonometric identities.

### Substitution and Elimination Method

However, instead of going deep into transforming these into a specific form, the approach seen in the image uses some substitution and direct solving.

### Steps Taken in the Image:

1. **Use of Triangle and Pythagoras' Theorem**: 
   They formed a right triangle to solve for \(\cos\theta\) and \(\sin\theta\) using the equation \( (8x+1)^2 = 12^2 + (4x-1)^2 \).

2. **Simplification and Solution**:
   This leads to solving for \( x \), giving the roots \( x = -2 \) and \( x = \frac{3}{2} \).

### Conclusion:
The image provided a process where the values of \( x \) were found using algebraic techniques combined with trigonometry.

### Do you have any specific questions or need further details?

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#### Related Questions:
1. What trigonometric identity is useful for transforming equations of the form \(\cos\theta + k\sin\theta = c\)?
2. How can the Pythagorean Theorem be used in solving trigonometric equations?
3. What is the significance of finding \(R\) in the identity \(R\cos(\theta - \alpha)\)?
4. How would you solve the system if the second equation was \( \cos\theta + x\sin\theta = 4 \)?
5. Can you derive \( \alpha \) from \(\tan^{-1}(8)\) in the solution above?

#### Tip:
When dealing with trigonometric systems, always consider whether a transformation or an identity could simplify the equations.

Explore how to solve the trigonometric equation cos(theta) + 8sin(theta) = 4 step by step. This problem involves trigonometric identities and the Pythagorean Theorem. Suitable for high school students in Grades 10-12.

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