For math in trig functions, use the graph determine the values of x for which sin(x)+cos(x)=-1 for the general solution 

To determine the values of \( x \) for which \( \sin(x) + \cos(x) = -1 \), we can use the following approach:

### Step 1: Simplify the equation
We start by rewriting the expression \( \sin(x) + \cos(x) \).

\[
\sin(x) + \cos(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)
\]

This comes from the identity:

\[
\sin(x) + \cos(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)
\]

### Step 2: Solve the equation
Now, we need to solve:

\[
\sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = -1
\]

Dividing both sides by \( \sqrt{2} \):

\[
\sin\left(x + \frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}
\]

### Step 3: Find the general solution
The sine function equals \( -\frac{\sqrt{2}}{2} \) at specific angles. The general solution for \( \sin(\theta) = -\frac{\sqrt{2}}{2} \) is:

\[
\theta = \frac{7\pi}{4} + 2n\pi \quad \text{or} \quad \theta = \frac{5\pi}{4} + 2n\pi \quad \text{where } n \in \mathbb{Z}
\]

Here, \( \theta = x + \frac{\pi}{4} \). So, substituting back:

\[
x + \frac{\pi}{4} = \frac{7\pi}{4} + 2n\pi \quad \text{or} \quad x + \frac{\pi}{4} = \frac{5\pi}{4} + 2n\pi
\]

### Step 4: Solve for \( x \)
Now, subtract \( \frac{\pi}{4} \) from both sides to solve for \( x \):

1. For \( \theta = \frac{7\pi}{4} \):

\[
x = \frac{7\pi}{4} - \frac{\pi}{4} + 2n\pi = \frac{6\pi}{4} + 2n\pi = \frac{3\pi}{2} + 2n\pi
\]

2. For \( \theta = \frac{5\pi}{4} \):

\[
x = \frac{5\pi}{4} - \frac{\pi}{4} + 2n\pi = \frac{4\pi}{4} + 2n\pi = \pi + 2n\pi
\]

### Final Answer
The general solution for \( x \) is:

\[
x = \frac

Explore how to solve the equation sin(x) + cos(x) = -1 using trigonometric identities and the sum-to-product formulas. This problem involves finding general solutions in trigonometry, suitable for advanced high school students.

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