Find the principal values of the following questions:

$c o s^{- 1} (- \frac{1}{2})$

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Solution

Let $c o s^{- 1} (- \frac{1}{2}) = θ \Rightarrow c o s θ = - \frac{1}{2}$
We know that the range of principal value of $c o s^{- 1} θ i s [0, π],$
$∴ c o s θ = - \frac{1}{2} = - c o s \frac{π}{3} = c o s (π - c o s \frac{π}{3}) (∵ c o s (π - θ) = - c o s θ)$
$= c o s \frac{2 π}{3}$

$\Rightarrow θ = \frac{2 π}{3}$ , where $θ ϵ [0, π] \Rightarrow c o s^{- 1} (- \frac{1}{2}) = \frac{2 π}{3}$
Hence, principal value of $c o s^{- 1} (- \frac{1}{2})$ is $c o s \frac{2 π}{3}$
Note $c o s^{- 1} (- θ) \neq - c o s^{- 1} θ$

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