The value of the determinant `|(sinA,cosA,sin(A+theta)),(sinB,cosB,sin(B+theta)),(sinC,cosC,sin(C+theta))|` is independent of

Options

A. A
B. B
C. C
D. `theta`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} \sin A & \cos A & \sin(A + \theta) \\ \sin B & \cos B & \sin(B + \theta) \\ \sin C & \cos C & \sin(C + \theta) \end{vmatrix} \]

we will perform a series of operations on the determinant to simplify it.

### Step 1: Apply Column Operations
We will first apply column operations to simplify the determinant. Specifically, we will multiply the first column \(C_1\) by \(\cos \theta\) and the second column \(C_2\) by \(\sin \theta\).

\[ D = \begin{vmatrix} \sin A \cos \theta & \cos A \sin \theta & \sin(A + \theta) \\ \sin B \cos \theta & \cos B \sin \theta & \sin(B + \theta) \\ \sin C \cos \theta & \cos C \sin \theta & \sin(C + \theta) \end{vmatrix} \]

### Step 2: Use the Sine Addition Formula
Next, we will use the sine addition formula:

\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]

Applying this to \(\sin(A + \theta)\), \(\sin(B + \theta)\), and \(\sin(C + \theta)\):

\[ \sin(A + \theta) = \sin A \cos \theta + \cos A \sin \theta \]
\[ \sin(B + \theta) = \sin B \cos \theta + \cos B \sin \theta \]
\[ \sin(C + \theta) = \sin C \cos \theta + \cos C \sin \theta \]

### Step 3: Substitute Back into the Determinant
Now we substitute these back into the determinant:

\[ D = \begin{vmatrix} \sin A \cos \theta & \cos A \sin \theta & \sin A \cos \theta + \cos A \sin \theta \\ \sin B \cos \theta & \cos B \sin \theta & \sin B \cos \theta + \cos B \sin \theta \\ \sin C \cos \theta & \cos C \sin \theta & \sin C \cos \theta + \cos C \sin \theta \end{vmatrix} \]

### Step 4: Simplify the Determinant
Now, we can see that the third column can be expressed as a linear combination of the first two columns. This means that the determinant will evaluate to zero because two columns are linearly dependent.

Thus, we conclude that:

\[ D = 0 \]

### Conclusion
The value of the determinant is independent of \(A\), \(B\), \(C\), and \(\theta\) since it evaluates to zero.

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