Question

Given that sin (α – β ) = $\frac{1}{2}$ and cos (α + β) = $\frac{1}{2}$, then α = _______ β = _________.

Answer 1

$$\begin{gathered} \sin \left(\alpha -\beta \right)=\frac{1}{2} \\ \Rightarrow \sin \left(\alpha -\beta \right)=\sin 30° \\ \Rightarrow \alpha -\beta =30°.....\left(1\right) \end{gathered}$$
Also,
$$\begin{gathered} \cos \left(\alpha +\beta \right)=\frac{1}{2} \\ \Rightarrow \cos \left(\alpha +\beta \right)=\cos 60° \\ \Rightarrow \alpha +\beta =60°.....\left(2\right) \end{gathered}$$
Adding (1) and (2), we get
$$\begin{gathered} \alpha -\beta +\alpha +\beta =30°+60° \\ \Rightarrow 2\alpha =90° \\ \Rightarrow \alpha =45° \end{gathered}$$
Putting $\alpha =45°$ in (2), we get
$$\begin{gathered} 45°+\beta =60° \\ \Rightarrow \beta =60°-45°=15° \end{gathered}$$
$\therefore \alpha =3\beta$

Given that sin (α – β ) = $\frac{1}{2}$ and cos (α + β) = $\frac{1}{2}$, then α = ___3___ β = ____45º___.