NIMCET 2013 — Mathematics PYQ

Step 1: Use the trigonometric formula for the area of a triangle

The area of a triangle when two sides and the included angle are given is:

Given:

• $\text{Area}=10\sqrt{3}$

• $c=|AB|=8$

• $b=|AC|=5$

Step 2: Substitute the values into the formula

Step 3: Solve for $\sin (A)$

Step 4: Determine the possible values for angle $A$

Since the sine function is positive in both the first and second quadrants, there are two possible angles for $A$ within the range of a triangle (0^\circ < A < 180^\circ):

• Case 1 (Acute): $A={\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=60^{\circ }$

• Case 2 (Obtuse): $A=180^{\circ }-60^{\circ }=120^{\circ }$

Both $60^{\circ }$ and $120^{\circ }$ are valid as they are less than $180^{\circ }$.

Correct Option: 1. $60^{\circ }$ or $120^{\circ }$