Determine the value of $sec( heta)$ given $cos( heta) = -frac{1}{sqrt{2}}$ and $ heta$ is in the third quadrant

Given that $\cos(\theta) = -\frac{1}{\sqrt{2}}$ and $\theta$ is in the third quadrant, we start by using the identity:

$\sec(\theta) = \frac{1}{\cos(\theta)}$

Substituting the given value:

$\sec(\theta) = \frac{1}{-\frac{1}{\sqrt{2}}}$

We simplify the fraction:

$\sec(\theta) = -\sqrt{2}$

Thus, the value of $\sec(\theta)$ is $-\sqrt{2}$.

Given $cos( heta) = -frac{1}{sqrt{2}}$ and $ heta$ in the third quadrant:

$sec( heta) = frac{1}{cos( heta)}$

$sec( heta) = frac{1}{-frac{1}{sqrt{2}}} = -sqrt{2}.$

Given $cos( heta) = -frac{1}{sqrt{2}}$:

$sec( heta) = -sqrt{2}.$

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