Find the principal values of the following questions:

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

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Solution

Let $c o s^{- 1} (- \frac{1}{\sqrt{2}}) = θ \Rightarrow c o s θ = - \frac{1}{\sqrt{2}}$
We know that the range of principal value of $c o s^{- 1} θ i s [0, π]$
$∴ c o s θ = - \frac{1}{\sqrt{2}} = - c o s \frac{π}{4} = c o s (π - \frac{π}{4}) [∵ c o s (π - θ) = - c o s θ]$
$\Rightarrow θ = \frac{3 π}{4}$ , where $θ ϵ [0, π] \Rightarrow c o s^{- 1} (- \frac{1}{\sqrt{2}}) = \frac{3 π}{4}$
Hence, the principal value of $c o s^{- 1} (- \frac{1}{\sqrt{2}}) i s \frac{3 π}{4}$

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