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Question

$${\text{The}}\;{\text{principal}}\;{\text{value}}\;{\text{of}}\;{\cos ^{ - 1}}\left( { - \sin \dfrac{{7\pi }}{6}} \right){\text{is}}\;{\text{:}}$$


A
5π3
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B
7π6
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C
π3
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D
none of these
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Solution

The correct option is B $$\frac{{5\pi }}{3}$$
We have, $$\cos ^{ -1 }{ \left( -\sin { \left( \dfrac { 7\pi  }{ 6 }  \right)  }  \right)  } $$
$$\Rightarrow \cos ^{ -1 }{ \left( \cos { \left( \dfrac { \pi  }{ 2 } +\dfrac { 7\pi  }{ 6 }  \right)  }  \right)  } $$
$$\Rightarrow \cos ^{ -1 }{ \left( \cos { \left( \dfrac { 10\pi  }{ 6 }  \right)  }  \right)  } $$
$$\Rightarrow \cos ^{ -1 }{ \left( \cos { \left( \dfrac { 5\pi  }{ 3 }  \right)  }  \right)  } $$
$$\Rightarrow \dfrac { 5\pi  }{ 3 } $$

Mathematics

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