First slide

Equation of line in Straight lines

Question

If the points $O (0, 0), A (\cos α, \sin α),$ $B (\cos β, \sin β)$ are the vertices of a right-angled triangle, then $|\sin (\frac{α - β}{2})| =$

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{\sqrt{2}}$
C

$\frac{1}{\sqrt{3}}$
D

none of these

Solution

We have $O A = O B = 1$
and $A B = \sqrt{2 - 2 \cos (α - β)} = 2 |\sin (\frac{α - β}{2})|$
Since $∆ A O B$ is a right triangle. Therefore
$\begin{array}{l} A B = \sqrt{O A^{2} + O B^{2}} \\ \Rightarrow 2 |\sin \frac{α - β}{2}| = \sqrt{2} \Rightarrow |\sin \frac{α - β}{2}| = \frac{1}{\sqrt{2}} \end{array}$

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