Use the given information about the angle $\theta$ to find the exact value of each of the following.
a. $\sin 2 \theta$
b. $\cos 2 \theta$
c. $\tan 2 \theta$
\[
\tan \theta=\frac{8}{7}, \sin \theta< 0
\]
a. $\sin 2 \theta=$
(Type an integer or a simplified fraction. Type an exact answer, using radicals as needed.)
b. $\cos 2 \theta=$
(Type an integer or a simplified fraction. Type an exact answer, using radicals as needed.)
c. $\tan 2 \theta=$
(Type an integer or a simplified fraction. Type an exact answer, using radicals as needed.)

Step 1 ：We are given that \(\tan \theta=\frac{8}{7}\) and \(\sin \theta<0\). We need to find the values of \(\sin 2 \theta\), \(\cos 2 \theta\) and \(\tan 2 \theta\).

Step 2 ：We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). From this, we can find the values of \(\sin \theta\) and \(\cos \theta\).

Step 3 ：Since \(\tan \theta=\frac{8}{7}\), we can consider this as a right triangle where the opposite side is 8 and the adjacent side is 7. Using the Pythagorean theorem, we can find the hypotenuse which is \(\sqrt{8^2 + 7^2} = \sqrt{113}\).

Step 4 ：So, \(\sin \theta = \frac{opposite}{hypotenuse} = \frac{8}{\sqrt{113}}\) and \(\cos \theta = \frac{adjacent}{hypotenuse} = \frac{7}{\sqrt{113}}\).

Step 5 ：But we are given that \(\sin \theta<0\), so \(\sin \theta = -\frac{8}{\sqrt{113}}\).

Step 6 ：Now, we can use the double angle formulas to find \(\sin 2 \theta\), \(\cos 2 \theta\) and \(\tan 2 \theta\).

Step 7 ：\(\sin 2 \theta = 2 \sin \theta \cos \theta\)

Step 8 ：\(\cos 2 \theta = \cos^2 \theta - \sin^2 \theta\)

Step 9 ：\(\tan 2 \theta = \frac{\sin 2 \theta}{\cos 2 \theta}\)

Step 10 ：Let's calculate these values.

Step 11 ：Final Answer: \(\sin 2 \theta = \boxed{-0.991}\)

Step 12 ：Final Answer: \(\cos 2 \theta = \boxed{-0.133}\)

Step 13 ：Final Answer: \(\tan 2 \theta = \boxed{7.467}\)