Question

If $\angle AOB=\theta$, then $\cos 2\theta$ is

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{6}$

Answer 1

The correct option is A $\frac{1}{8}$

$OABC$ (anti-clockwise) is a rhombus where $O$ is the origin, $A,B,C$ are represented by the complex numbers . $z_1,z_2$ and $z_{3.}$
$z_1$ and $z_3$ lie on $|z|=2.z_2$ lies on $|z|=3$.

Further $\arg (z_1-z_2)=\frac{\pi }{3}$ as $OABC$ is a rhombus all sides are equal.
i.e $|Z_1|=|Z_3|=|Z_1-Z_2|=2$ as $Z_1$ and $Z_3$ lie on $|Z|=2$
$\angle AOB=\theta$
in $\Delta AOB$ $OA=|Z_1|=2,AB=|Z_1-Z_2|=2,OB=|Z_2|=3$
Using cosine rule
$\cos \theta =\frac{O{A}^2+O{B}^2-A{B}^2}{2\cdot OA\cdot OB}=\frac{4+9-4}{12}$
$\Rightarrow \cos \theta =\frac{3}{4}$
$\therefore \cos 2\theta =2{\cos }^2\theta -1=\frac{1}{8}$
Hence, option A.