Discussion
The value of
\[ \begin{vmatrix} \sin 30^\circ & \cos 30^\circ & \sin(30^\circ + 75^\circ) \\ \sin 45^\circ & \cos 45^\circ & \sin(45^\circ + 75^\circ) \\ \sin 60^\circ & \cos 60^\circ & \sin(60^\circ + 75^\circ) \end{vmatrix} \]
is equal to
Answer: C
The determinant is 0.
Expand the third column with the sine addition formula: \(\sin(\theta + 75^\circ) = \sin\theta\cos 75^\circ + \cos\theta\sin 75^\circ\).
\[ C_3 = \cos 75^\circ \, C_1 + \sin 75^\circ \, C_2 \]
The third column is a linear combination of the first two (with the same constants \(\cos 75^\circ\) and \(\sin 75^\circ\) in every row), so the operation \(C_3 \rightarrow C_3 - \cos 75^\circ C_1 - \sin 75^\circ C_2\) makes it a column of zeros and the determinant vanishes. No numerical evaluation is needed.
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