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Question: If \(\sin(\alpha - \beta) = \frac{1}{2}\)and \(\cos(\alpha + \beta) = \frac{1}{2},\) where \(\alpha\...

If sin⁡(α−β)=12\sin(\alpha - \beta) = \frac{1}{2}and cos⁡(α+β)=12,\cos(\alpha + \beta) = \frac{1}{2}, where α\alpha and β\beta are positive acute angles, then

A

α=45∘,β=15∘\alpha = 45{^\circ},\beta = 15{^\circ}

B

α=15∘,β=45∘\alpha = 15{^\circ},\beta = 45{^\circ}

C

α=60∘,β=15∘\alpha = 60{^\circ},\beta = 15{^\circ}

D

None of these

Answer

α=45∘,β=15∘\alpha = 45{^\circ},\beta = 15{^\circ}

Explanation

Solution

sin⁡(α−β)=12=sin⁡30∘⇒α−β=30∘\sin(\alpha - \beta) = \frac{1}{2} = \sin 30{^\circ} \Rightarrow \alpha - \beta = 30{^\circ} …..(i)

and cos⁡(α+β)=12⇒α+β=60∘\cos(\alpha + \beta) = \frac{1}{2} \Rightarrow \alpha + \beta = 60{^\circ} …..(ii)

Solving (i) and (ii), we get α=45∘\alpha = 45{^\circ}and β=15∘\beta = 15{^\circ}.

Trick : In such type of problems, students should satisfy the given conditions with the values given in the options. Here α=45∘\alpha = 45{^\circ}and β=15∘\beta = 15{^\circ}satisfy both the conditions.