To solve the equation \(\sin x + \cos x\), let's approach it step by step:

1. **Square Both Sides**:
   \[
   (\sin x + \cos x)^2 = y^2
   \]
   This simplifies to:
   \[
   \sin^2 x + 2 \sin x \cos x + \cos^2 x = y^2
   \]

2. **Use Trigonometric Identities**:
   We know that \(\sin^2 x + \cos^2 x = 1\), so the equation becomes:
   \[
   1 + 2 \sin x \cos x = y^2
   \]

3. **Simplify the Equation**:
   \[
   y^2 = 1 + 2 \sin x \cos x
   \]

4. **Express \(2 \sin x \cos x\) in Terms of \(\sin 2x\)**:
   Using the double-angle identity for sine, we have:
   \[
   2 \sin x \cos x = \sin 2x
   \]
   Therefore, the equation becomes:
   \[
   y^2 = 1 + \sin 2x
   \]

To find specific values of \(x\) for which \(\sin x + \cos x = y\), we solve the original equation. The range of \(\sin x + \cos x\) is \([- \sqrt{2}, \sqrt{2}]\) due to the maximum and minimum values achievable by combining the sine and cosine functions. This implies that:

\[
-\sqrt{2} \leq y \leq \sqrt{2}
\]

If you are looking for specific solutions or if there are additional constraints, please provide more details.

Would you like further details or have any questions on this topic?

Here are 5 numbered questions you might find useful:
1. How do you solve trigonometric equations involving multiple angles?
2. What are the key trigonometric identities and how are they derived?
3. Can you explain the method to solve \(\sin x = \cos x\)?
4. How do you find the exact values of sine and cosine for common angles?
5. How does the unit circle relate to solving trigonometric equations?

**Tip:** Always remember to check the domain and range of trigonometric functions when solving equations to ensure the solutions are valid within the given context.

Learn how to solve the trigonometric equation sinx + cosx step-by-step with detailed explanations and the use of trigonometric identities. Suitable for advanced high school students and those preparing for trigonometry exams.

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