Evaluate the expression under the given conditions. \[ \cos (2 \theta) ; \quad \sin (\theta)=-\frac{4}{5}, \quad \theta \text { in Quadrant III } \]

Given that \(\sin(\theta) = -\frac{4}{5}\) and \(\theta\) is in Quadrant III, we can find the value of \(\cos(2\theta)\) using the double angle formula for cosine: \[ \cos(2\theta) = 1 - 2\sin^2(\theta) \] Substitute the given value of \(\sin(\theta)\) into the formula: \[ \cos(2\theta) = 1 - 2\left(-\frac{4}{5}\right)^2 \] Now, we can calculate the value of \(\cos(2\theta)\). Calculate the value by following steps: - step0: Calculate: \(1-2\left(\frac{-4}{5}\right)^{2}\) - step1: Rewrite the fraction: \(1-2\left(-\frac{4}{5}\right)^{2}\) - step2: Multiply the terms: \(1-\frac{32}{25}\) - step3: Reduce fractions to a common denominator: \(\frac{25}{25}-\frac{32}{25}\) - step4: Transform the expression: \(\frac{25-32}{25}\) - step5: Subtract the numbers: \(\frac{-7}{25}\) - step6: Rewrite the fraction: \(-\frac{7}{25}\) The value of \(\cos(2\theta)\) is \(-\frac{7}{25}\) or approximately \(-0.28\).