Answer
$2.42$
Work Step by Step
Let $\theta = \sec^{-1} {\left(-\frac{4}{3}\right)}$
Then, $\sec{\theta} = -\dfrac{4}{3}$.
Since $\cos{\theta}= \dfrac{1}{\sec{\theta}}$, then
$\cos{\theta} = \dfrac{1}{-\frac{4}{3}}=-\dfrac{3}{4}$
Thus,
$\theta = \cos^{-1} {\left(-\frac{3}{4}\right)}\approx 2.42$
Hence,
$\sec^{-1} {\left(-\frac{4}{3}\right)}= \cos^{-1} {\left(-\frac{3}{4}\right)} \approx \boxed{2.42}$
