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Find the value of the expression \( cos^{-1} \bigg(cos \frac{\large 7\pi}{\large 6} \bigg) \)

\[ \begin{array} (A) \frac{7\pi}{6} \quad & (B) \frac{5\pi}{6} \quad & (C) \frac{\pi}{3} \quad & (D) \frac{\pi}{6} \end{array} \]
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Toolbox:
  • cos(\(\pi+x)=-cosx\)
  • \(cos(\pi+x)=cos(\pi-x)\)
  • Principal interval of cos is [0,\(\pi\)]
Given $cos^{-1} \bigg(cos \frac{\large 7\pi}{\large 6} \bigg)$:
$\frac{7\pi}{6} = \pi+\frac{\pi}{6},\;$ However, \(\pi+\frac{\pi}{6}\) is not in the principal interval.
We know that \(cos(\pi+x)=cos(\pi-x)\). Therefore by taking $x=$\(\frac{\pi}{6}\) we can write
\(cos(\pi+\frac{\pi}{6})=cos(\pi-\frac{\pi}{6})\)
The expression then reduces to \(cos^{-1} \bigg[ cos \bigg( \pi-\frac{\pi}{6} \bigg) \bigg] = cos^{-1}cos \frac{5\pi}{6}\)
\( = \frac{5\pi}{6}\)
answered Feb 23, 2013 by thanvigandhi_1
edited Mar 14, 2013 by balaji.thirumalai
 

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