NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

In $\triangle ABC$ , if $a = 2$ , $b = 4$ and $\angle C = 60^{\circ}$ , then $A$ and $B$ are respectively equal to:

Choose the correct answer:

Explanation

Concept

For a triangle:

Cosine rule: $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab}$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Where $A$ , $B$ and $C$ are the angles of the triangle and $a$ , $b$ and $c$ are sides opposite to angle $A$ , $B$ and $C$ respectively.

Calculation

Given $a = 2$ , $b = 4$ and $\angle C = 60^{\circ}$ .

1. Find side $c$ using the Cosine rule:

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab}

\Rightarrow \cos 60^{\circ} = \frac{2^{2} + 4^{2} - c^{2}}{2(2 \times 4)}

\Rightarrow \frac{1}{2} = \frac{4 + 16 - c^{2}}{16}

\Rightarrow 8 = 20 - c^{2}

\Rightarrow c = \sqrt{12} = 2\sqrt{3}

2. Find angle $A$ using the Sine rule:

\frac{a}{\sin A} = \frac{c}{\sin C}

\Rightarrow \sin A = \frac{a \times \sin C}{c}

\Rightarrow \sin A = \frac{2 \times \sin 60^{\circ}}{2\sqrt{3}} = \frac{1}{2}

\Rightarrow \angle A = 30^{\circ}

3. Find angle $B$ using the Angle Sum Property:

\angle A + \angle B + \angle C = 180^{\circ}

\Rightarrow 30^{\circ} + B + 60^{\circ} = 180^{\circ}

\Rightarrow \angle B = 90^{\circ}

Correct Option: 4