Courses
Courses for Kids
Free study material
Free LIVE classes
More
LIVE
Join Vedantu’s FREE Mastercalss

If $A + B + C = \pi ,$then $\left| {\begin{array}{*{20}{c}}
  {\sin (A + B + C)}&{\sin B}&{\cos C} \\
  {\sin B}&0&{\tan A} \\
  {\cos (A + B)}&{ - \tan A}&0
\end{array}} \right| = .......................$

Answer
VerifiedVerified
366.6k+ views
Hint: $\sin \pi = 0$, $\cos (\pi - x) = - \cos x$
Using the properties of trigonometry first, we make the determinant simple and then further solve it.
Also given
$A + B + C = \pi$
Applying trigonometric properties, we get
$
  \sin (A + B + C) = \sin \pi = 0 \\
  \cos (A + B) = \cos (\pi - C) = - \cos C \\
$
Putting these values in the determinant, we get
$
  \Delta = \left| {\begin{array}{*{20}{c}}
  {\sin (A + B + C)}&{\sin B}&{\cos C} \\
  {\sin B}&0&{\tan A} \\
  {\cos (A + B)}&{ - \tan A}&0
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
  0&{\sin B}&{\cos C} \\
  {\sin B}&0&{\tan A} \\
  { - \cos C}&{ - \tan A}&0
\end{array}} \right| \\
    \\
$
Now expanding the determinant through row${R_1}$,we get
$\begin{gathered}
  \Delta = 0\left| {\begin{array}{*{20}{c}}
  0&{\tan A} \\
  { - \tan A}&0
\end{array}} \right| + ( - \sin B)\left| {\begin{array}{*{20}{c}}
  {\sin B}&{\tan A} \\
  { - \cos C}&0
\end{array}} \right| + \cos C\left| {\begin{array}{*{20}{c}}
  {\sin B}&0 \\
  { - \cos C}&{ - \tan A}
\end{array}} \right| \\
  \Delta = - \sin B(0 - (\tan A)( - \cos C)) + \cos C((\sin B)( - \tan A) - 0) \\
  \Delta = \sin B\tan A\cos C - \sin B\tan A\cos C \\
  \Delta = 0 \\
\end{gathered} $
Therefore, value of determinant,
$\Delta = \left| {\begin{array}{*{20}{c}}
  {\sin (A + B + C)}&{\sin B}&{\cos C} \\
  {\sin B}&0&{\tan A} \\
  {\cos (A + B)}&{ - \tan A}&0
\end{array}} \right| = 0$
Note: Use of trigonometric properties transforms complex determinant into simple determinant
So, first apply trigonometric properties and then expand the determinant. As this will be the
easiest and efficient way to get the solution of such problems.

Last updated date: 01st Oct 2023
Total views: 366.6k
Views today: 8.66k