Question

If $\cos 40^{\circ }=x$ and $\cos \theta =1-2x^2$, 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 $260^{\circ }$