Question

if $ \cos {{20}^{o}}=k $ and $ \cos x=2{{k}^{2}}-1, $ then the possible values of $ x $ between $ {{0}^{o}} $ and $ {{360}^{o}} $ are:

• A. $140^{\circ}$ and $270^{\circ}$
• B. $40^{\circ}$ and $140^{\circ}$
• C. $40^{\circ}$ and $320^{\circ}$
• D. $50^{\circ}$ and $130^{\circ}$

Answer 1

Answer: (C)

We have $k=\cos 20^{\circ}\,\,\,...$(i)
and $2 k^{2}-1=\cos x\,\,\,...$ (ii)
From Eqs.(i) and (ii), we get
$2 \cos ^{2} 20^{\circ}-1=\cos x$
$\Rightarrow \cos x=\cos 40^{\circ}$
$\Rightarrow x=40^{\circ}$
or $x=360^{\circ}-40^{\circ}=320^{\circ}$
$\therefore $ The values of $x$ lying between $0^{\circ}$ and $360^{\circ}$ are $40^{\circ}$ and $320^{\circ}$.