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The value of \(cosec^{-1}\left(cosec\frac{4\pi}{3}\right)\) isA. \(\frac{\pi}{3}\)B. \(\frac{-\pi}{3}\)C. \(\frac{2\pi}{3}\)D. none of these |
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Answer» Correct Answer is \(\frac{-\pi}{3}\) Now, let x = \(cosec^-1\left(cosec\frac{4\pi}{3}\right)\) ⇒ cosec x =cosec ( \(\frac{4\pi}{3}\)) Here range of principle value of cosec is [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] ⇒ x =\(\frac{4\pi}{3}\)∉ [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] Hence for all values of x in range [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] ,the value of \(cosec^-1\left(cosec\frac{4\pi}{3}\right)\) is ⇒ cosec x =cosec (π+ \(\frac{\pi}{3}\)) (\(\because\) cosec ( \(\frac{4\pi}{3}\))= cosec ( π+ \(\frac{\pi}{3}\)) ) ⇒ cosec x =cosec (-\(\frac{\pi}{3}\)) (\(\because\) cosec (π + θ)= cosec(-θ)) ⇒ x = -\(\frac{\pi}{3}\) |
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