The maximum value of 

y equals sine xy=sinx​, 

0less than or equals≤xless than or equals≤2piπ​, 

is​ _______ and occurs at 

xequals=​_______.


To find the maximum value of the function \(y = \sin x\) in the interval \(0 \leq x \leq 2\pi\), we need to analyze the sine function's behavior within this range.

### Step-by-Step Solution:

1. **Understanding the Sine Function**:
   - The sine function, \(\sin x\), oscillates between \(-1\) and \(1\).
   - Its maximum value is \(1\) and minimum value is \(-1\).

2. **Identifying the Maximum Value**:
   - The maximum value of \(\sin x\) is \(1\).

3. **Finding When the Maximum Occurs**:
   - The sine function reaches its maximum value of \(1\) at \(x = \frac{\pi}{2}\) within the interval \(0 \leq x \leq 2\pi\).

### Conclusion:

- The maximum value of \(y = \sin x\) is **1**.
- This maximum value occurs at \(x = \frac{\pi}{2}\).

So, the maximum value is **1**, and it occurs at \(x = \frac{\pi}{2}\).

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. What are the other key points (like zeros and minimums) of the sine function in the interval \(0 \leq x \leq 2\pi\)?
2. How does the function \(y = \cos x\) behave in the same interval?
3. If we shift the sine function by \(\frac{\pi}{4}\), how does that affect the maximum point?
4. What is the period of the sine function, and how does it relate to finding maximum and minimum values?
5. Can we generalize the points where \(\sin x = 1\) for any interval?

**Tip:** When dealing with trigonometric functions, remember to consider their periodic nature, which can help in identifying multiple points of interest over a range.

Explore how to find the maximum value of y = sin x within the interval 0 ≤ x ≤ 2π. Understand the behavior of the sine function, its maximum value, and when it occurs. Suitable for high school students studying trigonometry.

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