`sin x + cos (x + 30^(@)) = 0, 0 ^(@) lt x lt 360^(@)`

Answer

To solve the equation \( \sin x + \cos(x + 30^\circ) = 0 \) for \( 0^\circ < x < 360^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ \sin x + \cos(x + 30^\circ) = 0 \]

### Step 2: Use the cosine addition formula
We can expand \( \cos(x + 30^\circ) \) using the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
Here, \( a = x \) and \( b = 30^\circ \). Thus:
\[ \cos(x + 30^\circ) = \cos x \cos 30^\circ - \sin x \sin 30^\circ \]
Substituting the values of \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \):
\[ \cos(x + 30^\circ) = \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2} \]

### Step 3: Substitute back into the equation
Now substitute this back into the equation:
\[ \sin x + \left( \cos x \cdot \frac{\sqrt{3}}{2} - \sin x \cdot \frac{1}{2} \right) = 0 \]
This simplifies to:
\[ \sin x + \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x = 0 \]

### Step 4: Combine like terms
Combine the sine terms:
\[ \left(1 - \frac{1}{2}\right) \sin x + \frac{\sqrt{3}}{2} \cos x = 0 \]
This simplifies to:
\[ \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x = 0 \]

### Step 5: Multiply through by 2
To eliminate the fraction, multiply the entire equation by 2:
\[ \sin x + \sqrt{3} \cos x = 0 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ \sin x = -\sqrt{3} \cos x \]

### Step 7: Divide by cos x
Dividing both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)):
\[ \tan x = -\sqrt{3} \]

### Step 8: Find the general solutions
The tangent function is negative in the second and fourth quadrants. The reference angle for \( \tan^{-1}(\sqrt{3}) \) is \( 60^\circ \) (or \( \frac{\pi}{3} \) radians). Thus, the solutions for \( x \) are:
\[ x = 180^\circ - 60^\circ = 120^\circ \quad \text{(2nd quadrant)} \]
\[ x = 360^\circ - 60^\circ = 300^\circ \quad \text{(4th quadrant)} \]

### Final Answers
The solutions in the interval \( 0^\circ < x < 360^\circ \) are:
\[ x = 120^\circ, \quad x = 300^\circ \]