To determine the values of $\sin 2x$, $\cos 2x$, and $\tan 2x$, let's use the double angle identities:

$\sin 2x = 2 \sin x \cos x$ $\cos 2x = \cos^2 x - \sin^2 x$ $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$

a) $\sin x = \frac{3}{5}$ and $x$ is acute

To find $\cos x$, use the Pythagorean identity:

$\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5}$

Now, calculate:

$\sin 2x = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$

$\cos 2x = \left(\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}$

$\tan x = \frac{\sin x}{\cos x} = \frac{3/5}{4/5} = \frac{3}{4}$

$\tan 2x = \frac{2 \times \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}} = \frac{24}{7}$

b) $\cot x = -\frac{7}{24}$ and $x$ is obtuse

$\tan x = -\frac{24}{7}$

For $\tan x = \frac{\sin x}{\cos x}$, find $\sin x$ and $\cos x$. Assume positive signs as required.

Let $\sin x = \frac{k}{\sqrt{k^2 + 24^2}}$ where $\cos x = -\frac{24}{\sqrt{k^2 + 24^2}}$.

Note $x$ is obtuse, so $\sin x$ is positive and $\cos x$ is negative.

$\sin 2x = 2\sin x\cos x = 2 \times \frac{k}{\sqrt{k^2 + 24^2}} \times \left(-\frac{24}{\sqrt{k^2 + 24^2}}\right)$

$= \frac{-48k}{k^2 + 24^2}$

Use:

$\cos 2x = 1 - 2\sin^2 x$ $\tan 2x = -\frac{2 \times \frac{-24}{7}}{1 - \left( -\frac{24}{7} \right)^2}$

c) $\cos x = \frac{12}{13}$ and $\frac{3\pi}{2} \le x \le 2\pi$

Calculate $\sin x$:

$\sin x = -\sqrt{1 - \cos^2 x} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} = -\frac{5}{13}$

(This is negative because $\frac{3\pi}{2} \le x \le 2\pi$.)

$\sin 2x = 2 \times \frac{12}{13} \times -\frac{5}{13} = -\frac{120}{169}$

$\cos 2x = \left(\frac{12}{13}\right)^2 - \left(-\frac{5}{13}\right)^2 = \frac{144}{169} - \frac{25}{169} = \frac{119}{169}$

$\tan x = \frac{\sin x}{\cos x} = \frac{-\frac{5}{13}}{\frac{12}{13}} = -\frac{5}{12}$

$\tan 2x = \frac{2 \times -\frac{5}{12}}{1 - \left(-\frac{5}{12}\right)^2} = -\frac{120}{119}$

These solutions should help in understanding the calculations for different angle scenarios.