52) Consider the equation sin(x) = 2 - k , with 0 <= x <= pi This equation has at least one solution if and only if: a) k >= 1 b) 1 <= k <= 2 c) k <= 2 d) - 1 <= k <= 1 e) 1 <= k <= 3

The equation to consider is \( \sin(x) = 2 - k \) with \( 0 \leq x \leq \pi \).

### Step 1: Understand the range of \(\sin(x)\)
The sine function, \(\sin(x)\), for \( x \) in the interval \([0, \pi]\), takes values in the range \([0, 1]\).

### Step 2: Analyze the equation \( \sin(x) = 2 - k \)
For the equation \( \sin(x) = 2 - k \) to have a solution within this interval, the right-hand side, \( 2 - k \), must be within the range \([0, 1]\).

This leads to the inequalities:
\[
0 \leq 2 - k \leq 1
\]

### Step 3: Solve the inequalities
1. For \( 0 \leq 2 - k \):
   \[
   k \leq 2
   \]
   
2. For \( 2 - k \leq 1 \):
   \[
   k \geq 1
   \]

### Step 4: Combine the conditions
Combining the results from the two inequalities, we get:
\[
1 \leq k \leq 2
\]

### Conclusion
The equation \( \sin(x) = 2 - k \) has at least one solution within the interval \( 0 \leq x \leq \pi \) if and only if \( 1 \leq k \leq 2 \).

The correct answer is:
**(b) \( 1 \leq k \leq 2 \)**

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Would you like further details or have any questions? Here are some related questions:

1. What is the general range of the sine function over different intervals?
2. How does the sine function behave in different quadrants of the unit circle?
3. What are the conditions for trigonometric equations to have solutions within specific intervals?
4. How can we solve inequalities involving trigonometric functions?
5. What other types of equations can involve the sine function and similar conditions?

**Tip:** When solving trigonometric equations, always consider the range of the trigonometric function over the specified interval.

Explore how to solve the trigonometric equation sin(x) = 2 - k over the interval [0, π]. Understand the range of the sine function and analyze inequalities to determine the conditions for solution. Suitable for high school students studying trigonometry and inequalities.

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