$2 \theta$ , given $\cos \theta=\frac{-12}{13}$ and $\sin \theta>0$

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Evaluate expression when two complex numbers are given

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2 \theta

, given

\cos \theta=\frac{-12}{13}

and

\sin \theta>0

Given information: We are given that $\cos \theta = \frac{-12}{13}$ and $\sin \theta > 0$ . We need to find the value of $2\theta$ . To find $2\theta$ , we first need to find the value of $\theta$ . Since $\cos \theta$ is negative and $\sin \theta$ is positive, $\theta$ must be in the second quadrant.
Find $\theta$ : In the second quadrant, the sine function is positive, which is consistent with the given information that $\sin \theta > 0$ . We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$ . $\sin^2 \theta = 1 - \cos^2 \theta$ $\sin^2 \theta = 1 - \left(\frac{-12}{13}\right)^2$
Calculate $\sin \theta$ : Calculate $\sin^2 \theta$ . $\newline$ $\sin^2 \theta = 1 - (\frac{144}{169})$ $\newline$ $\sin^2 \theta = (\frac{169}{169}) - (\frac{144}{169})$ $\newline$ $\sin^2 \theta = \frac{25}{169}$
Determine quadrant: Since $\sin \theta$ is positive in the second quadrant, we take the positive square root of $\sin^2 \theta$ to find $\sin \theta$ . $\newline$ $\sin \theta = \sqrt{\frac{25}{169}}$ $\newline$ $\sin \theta = \frac{5}{13}$
Use angle addition formula: Now we have both $\sin \theta$ and $\cos \theta$ . We can use the angle addition formula for cosine to find $\cos(2\theta)$ : $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ $\cos(2\theta) = \left(\frac{-12}{13}\right)^2 - \left(\frac{5}{13}\right)^2$
Calculate $\cos(2\theta)$ : Calculate $\cos(2\theta)$ . $\newline$ $\cos(2\theta) = \frac{144}{169} - \frac{25}{169}$ $\newline$ $\cos(2\theta) = \frac{144 - 25}{169}$ $\newline$ $\cos(2\theta) = \frac{119}{169}$
Determine quadrant for $2\theta$ : To find $2\theta$ , we need to determine the angle whose cosine is $119/169$ . Since $\cos(2\theta)$ is positive, $2\theta$ could be in the first or fourth quadrant. However, since $\theta$ is in the second quadrant, $2\theta$ must be in the first quadrant (as doubling an angle in the second quadrant would place it in the first quadrant). Therefore, we find the angle whose cosine is $119/169$ in the first quadrant.
Use inverse cosine function: We use the inverse cosine function to find $2\theta$ . $\newline$ $2\theta = \cos^{-1}(\frac{119}{169})$ $\newline$ This would typically require a calculator to find an approximate value, as $\frac{119}{169}$ is not a standard angle.