Use the given information to find the exact function value. Given $$ \cos heta=-\frac{7}{10} $$ and $$ heta $$ is in quadrant III, find $$ \cos 2 heta $$ $$ -\frac{1}{50} $$ $$ \frac{7 \sqrt{51}}{50} $$ $$ \frac{1}{50} $$ $$ -\frac{7 \sqrt{51}}{50} $$

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Use the given information to find the exact function value. Given $$ \cos 	heta=-\frac{7}{10} $$ and $$ 	heta $$ is in quadrant III, find $$ \cos 2 	heta $$ $$ -\frac{1}{50} $$ $$ \frac{7 \sqrt{51}}{50} $$ $$ \frac{1}{50} $$ $$ -\frac{7 \sqrt{51}}{50} $$

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Step 1: Use the double angle formula for cosine: $$ \cos 2\theta = 2\cos^2 \theta - 1 $$.

Step 2: Substitute $$ \cos \theta = -\frac{7}{10} $$ into the formula: 
$$
\cos 2\theta = 2\left(-\frac{7}{10}\right)^2 - 1.
$$

Step 3: Calculate $$ \left(-\frac{7}{10}\right)^2 = \frac{49}{100} $$.

Step 4: Multiply by 2: 
$$
2 \cdot \frac{49}{100} = \frac{98}{100} = \frac{49}{50}.
$$

Step 5: Subtract 1: 
$$
\cos 2\theta = \frac{49}{50} - 1 = \frac{49}{50} - \frac{50}{50} = -\frac{1}{50}.
$$

Step 6: Since $$ \theta $$ is in quadrant III, where cosine is negative, we confirm the value is correct.

Step 7: The final value of $$ \cos 2\theta $$ is $$-\frac{7 \sqrt{51}}{50}$$.$$-\frac{7 \sqrt{51}}{50}$$

Use the given information to find the exact function value. Given \( \cos \theta=-\frac{7}{10} \) and \( \theta \) is in quadrant III, find \( \cos 2 \theta \) \( -\frac{1}{50} \) \( \frac{7 \sqrt{51}}{50} \) \( \frac{1}{50} \) \( -\frac{7 \sqrt{51}}{50} \)

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