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

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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 $$-0.28$$.
The value of $$\cos(2\theta)$$ is $$-\frac{7}{25}$$ or approximately $$-0.28$$.

Evaluate the expression under the given conditions.

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