Question

If two sides of a triangle be given, then its area will be maximum if the angle between the given sides be-

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is C $\frac{\pi }{2}$

Let the sides be $a,b,c$ and angles be $A,B,C$.
Hence by solution of triangles.
$Area$
$=\frac{ab.\sin C}{2}$
Now for fixed values of $a$ and $b$, area is maximum if $\sin C=1$.
Or
$\sin C=\sin \frac{\pi }{2}$
Or
$C=\frac{\pi }{2}$.
Hence the area will be maximum if the angle between the given sides be ${90}^0$.