find all values on the graph of f(x) = x -3cos(x) where the tangent line has slope 1

To find the points on the graph of the function \( f(x) = x - 3\cos(x) \) where the tangent line has a slope of 1, we need to determine where the derivative of the function is equal to 1.

### Steps:

1. **Find the derivative of the function**:  
   The derivative \( f'(x) \) represents the slope of the tangent line at any point \( x \) on the curve.

   \[
   f'(x) = \frac{d}{dx}(x - 3\cos(x)) = 1 + 3\sin(x)
   \]

2. **Set the derivative equal to 1**:  
   Since we are looking for where the slope of the tangent line is 1, we set \( f'(x) = 1 \).

   \[
   1 + 3\sin(x) = 1
   \]

3. **Solve for \( \sin(x) \)**:  
   Simplifying the equation:

   \[
   3\sin(x) = 0
   \]

   \[
   \sin(x) = 0
   \]

4. **Find the values of \( x \)**:  
   The sine function equals 0 at integer multiples of \( \pi \):

   \[
   x = n\pi \quad \text{where} \, n \, \text{is an integer}
   \]

5. **Find the corresponding points**:  
   We now substitute these values of \( x \) back into the original function to find the corresponding \( y \)-coordinates:

   \[
   f(n\pi) = n\pi - 3\cos(n\pi)
   \]

   Since \( \cos(n\pi) = (-1)^n \), we get:

   \[
   f(n\pi) = n\pi - 3(-1)^n
   \]

   Thus, the points on the graph where the tangent line has slope 1 are \( \left( n\pi, n\pi - 3(-1)^n \right) \) for integer values of \( n \).

### Summary:

The tangent line to the graph of \( f(x) = x - 3\cos(x) \) has a slope of 1 at the points \( \left( n\pi, n\pi - 3(-1)^n \right) \), where \( n \) is any integer.

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Would you like more details or have any further questions?

### Relative Questions:
1. What happens when the slope is not 1 but some other value?
2. How does the function behave for large values of \( n \)?
3. What is the second derivative of the function, and what does it indicate about concavity at these points?
4. How would the graph of \( f(x) = x - 3\cos(x) \) differ from \( f(x) = x \)?
5. Can you find the intercepts of the function \( f(x) = x - 3\cos(x) \)?

### Tip:
When solving trigonometric equations like \( \sin(x) = 0 \), remember that the sine function has periodic solutions at multiples of \( \pi \).

Find all values on the graph of f(x) = x - 3cos(x) where the tangent line has slope 1.

Explore how to find points on the graph of f(x) = x - 3cos(x) where the tangent line has a slope of 1. This detailed solution involves concepts of differentiation and trigonometry, suitable for high school students in Grades 10-12.

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