Question

If $\cos 40^{\circ}=x$ and $\cos \theta=1-2 x^{2}$, then the possible values of $\theta$ lying between $0^{\circ}$ and $360^{\circ}$ are

• A. $100^{\circ}$ and $260^{\circ}$
• B. $110^{\circ}$ and $280^{\circ}$
• C. $80^{\circ}$ and $280^{\circ}$
• D. $110^{\circ}$ and $260^{\circ}$

Answer 1

Answer: (B)

Here, $\cos \theta=1-2 \cos ^{2} 40^{\circ}$
$=-\left(2 \cos ^{2} 40^{\circ}-1\right)$
$=-\cos \left(2 \times 40^{\circ}\right)=-\cos 80^{\circ}$
So, $\cos \theta=\cos \left(180^{\circ}+80^{\circ}\right)$ or $\cos \theta=\cos \left(180^{\circ}-80^{\circ}\right)$
Hence, $\theta=100^{\circ}$ and $260^{\circ}$