MHT CET · Maths · Trigonometric Ratios & Identities
The principal solutions, of the equation \(\sqrt{3} \sec x+2=0\), are
- A \(\frac{2 \pi}{3}, \frac{4 \pi}{3}\)
- B \(\frac{4 \pi}{3}, \frac{5 \pi}{3}\)
- C \(\frac{5 \pi}{6}, \frac{7 \pi}{6}\)
- D \(\frac{7 \pi}{6}, \frac{11 \pi}{6}\)
Answer & Solution
Correct Answer
(C) \(\frac{5 \pi}{6}, \frac{7 \pi}{6}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& \sqrt{3} \sec x+2=0 \\
& \Rightarrow \sec x=\frac{-2}{\sqrt{3}} \\
& \Rightarrow \cos x=\frac{-\sqrt{3}}{2} \\
& \Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{6}\right)=\cos \frac{5 \pi}{6} \\
& \text { and } \cos x=\cos \left(\pi+\frac{\pi}{6}\right)=\cos \frac{7 \pi}{6}
\end{aligned}\)
\(\therefore\) The principal solutions are \(\frac{5 \pi}{6}\) and \(\frac{7 \pi}{6}\)
& \sqrt{3} \sec x+2=0 \\
& \Rightarrow \sec x=\frac{-2}{\sqrt{3}} \\
& \Rightarrow \cos x=\frac{-\sqrt{3}}{2} \\
& \Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{6}\right)=\cos \frac{5 \pi}{6} \\
& \text { and } \cos x=\cos \left(\pi+\frac{\pi}{6}\right)=\cos \frac{7 \pi}{6}
\end{aligned}\)
\(\therefore\) The principal solutions are \(\frac{5 \pi}{6}\) and \(\frac{7 \pi}{6}\)
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