If `sin theta_1+sin theta_2+sin theta_3+sin theta_4=4`, then what is the value of `cos theta_1+cos theta_2+cos theta_3+cos theta_4=?`

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we start with the equation given:

\[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 + \sin \theta_4 = 4 \]

### Step 1: Analyze the sine function
The sine function, \(\sin \theta\), has a maximum value of 1 for any angle \(\theta\). Therefore, for the equation \(\sin \theta_1 + \sin \theta_2 + \sin \theta_3 + \sin \theta_4\) to equal 4, each sine term must be at its maximum value.

### Step 2: Set each sine term to its maximum
This means:

\[ \sin \theta_1 = 1, \quad \sin \theta_2 = 1, \quad \sin \theta_3 = 1, \quad \sin \theta_4 = 1 \]

### Step 3: Determine the angles
The angles that satisfy \(\sin \theta = 1\) are:

\[ \theta_1 = 90^\circ + 360^\circ k, \quad \theta_2 = 90^\circ + 360^\circ k, \quad \theta_3 = 90^\circ + 360^\circ k, \quad \theta_4 = 90^\circ + 360^\circ k \]

for any integer \(k\).

### Step 4: Calculate the cosine values
Now, we need to find the cosine values for these angles:

\[ \cos \theta_1 = \cos(90^\circ) = 0 \]
\[ \cos \theta_2 = \cos(90^\circ) = 0 \]
\[ \cos \theta_3 = \cos(90^\circ) = 0 \]
\[ \cos \theta_4 = \cos(90^\circ) = 0 \]

### Step 5: Sum the cosine values
Now, we sum the cosine values:

\[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 + \cos \theta_4 = 0 + 0 + 0 + 0 = 0 \]

### Conclusion
Thus, the value of \(\cos \theta_1 + \cos \theta_2 + \cos \theta_3 + \cos \theta_4\) is:

\[ \boxed{0} \]