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

Question

If two sides of a triangle are given, then the area of the triangle will be maximum, if the angle between the given sides is (A) (π/3) (B) (π/4) (C) (π/6) (D) (π/2)

Tardigrade Admin · Accepted Answer

Let a and b be given, then area, A=(1/2) a b sin C ⇒ (d A/d C)=(1/2) a b cos C <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/e1b8d07858906d36e9d000d2458e8fa7-.png" /> Hence, A is maximum, when (d A/d C)=0 ⇒ C=90°=(π/2)

Q. If two sides of a triangle are given, then the area of the triangle will be maximum, if the angle between the given sides is

$\frac{π}{3}$

$\frac{π}{4}$

$\frac{π}{6}$

$\frac{π}{2}$

Let $a$ and $b$ be given, then area, $A = \frac{1}{2} ab sin C$
$\Rightarrow \frac{d A}{d C} = \frac{1}{2} ab cos C$

Hence, $A$ is maximum, when $\frac{d A}{d C} = 0$
$\Rightarrow C = 9 0^{\circ} = \frac{π}{2}$