`sin(180+theta)+cos(270+theta)+cos(90+theta)+sin*(360+theta)` = __________.

Answer

To solve the expression \( \sin(180^\circ + \theta) + \cos(270^\circ + \theta) + \cos(90^\circ + \theta) + \sin(360^\circ + \theta) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \sin(180^\circ + \theta) \)
Using the sine addition formula, we know:
\[ \sin(180^\circ + \theta) = -\sin(\theta) \]
This is because sine is negative in the third quadrant.

### Step 2: Evaluate \( \cos(270^\circ + \theta) \)
Using the cosine addition formula, we have:
\[ \cos(270^\circ + \theta) = -\sin(\theta) \]
This is because cosine is negative in the fourth quadrant.

### Step 3: Evaluate \( \cos(90^\circ + \theta) \)
Using the cosine addition formula, we find:
\[ \cos(90^\circ + \theta) = -\sin(\theta) \]
This is because cosine is negative in the second quadrant.

### Step 4: Evaluate \( \sin(360^\circ + \theta) \)
Using the sine addition formula, we have:
\[ \sin(360^\circ + \theta) = \sin(\theta) \]
This is because sine is positive in the first quadrant.

### Step 5: Combine all the results
Now, we can substitute all the evaluated terms back into the original expression:
\[ \sin(180^\circ + \theta) + \cos(270^\circ + \theta) + \cos(90^\circ + \theta) + \sin(360^\circ + \theta) \]
Substituting the values we found:
\[ -\sin(\theta) + (-\sin(\theta)) + (-\sin(\theta)) + \sin(\theta) \]
This simplifies to:
\[ -\sin(\theta) - \sin(\theta) - \sin(\theta) + \sin(\theta) = -2\sin(\theta) + \sin(\theta) = -\sin(\theta) \]

### Final Result
Thus, the final result is:
\[ -\sin(\theta) \]