Question

Sol. [Key Concept: Right angle at one vertex of $A(0,0),B(\cos\ \alpha,\sin\ \alpha),C(\cos\ \beta,\sin\ \beta)$ ⇒ conditions on $\frac{\alpha - \beta}{2}$.]

$$\begin{matrix} \alpha - \beta & \ = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \\ \frac{\alpha - \beta}{2} & \ = \pm \frac{\pi}{4}, \pm \frac{3\pi}{4} \\ cos\left( \frac{\alpha - \beta}{2} \right) & \ = \frac{1}{\sqrt{2}}, - \frac{1}{\sqrt{2}} \\ sin\left( \frac{\alpha - \beta}{2} \right) & \ = \pm \frac{1}{\sqrt{2}} \end{matrix}$$

✔ (a) $\sin\frac{\alpha - \beta}{2} = \frac{12}{}$ gives a right angle.

✔ (b) $\cos\frac{\alpha - \beta}{2} = - \frac{12}{}$ gives a right angle.

✔ (c) $\cos\frac{\alpha - \beta}{2} = \frac{12}{}$ gives a right angle.

✔ (d) $\sin\frac{\alpha - \beta}{2} = - \frac{12}{}$ gives a right angle.