Matrices & Determinants
Determinants with Trigonometric Functions
Grade 12
Question:
<p>If \(A\), \(B\) and \(C\) are the angles of a non-right angled triangle \(ABC\), the value of <span style='display:inline-block; border: 1px solid black; padding: 5px;'>\[\begin{vmatrix} \tan A & 1 & 1 \\ 1 & \tan B & 1 \\ 1 & 1 & \tan C \end{vmatrix}\]</span> is</p>
<p>(a) \(0\)</p>
<p>(b) \(1\)</p>
<p>(c) \(2\)</p>
<p>(d) \(3\)</p>
Step-by-Step Solution
Key Concept: For a triangle with angles summing to $\pi$, there is a special identity relating the tangents, which causes the determinant to vanish.
<p>Use the property that $A + B + C = \pi$ in a triangle. Expand the determinant and apply trigonometric identities such as $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ (for non-right triangles). This simplifies the determinant to zero.</p>
Correct Answer: A