(Calculus) When $\sin \alpha = \frac { 3 } { 4 }$, what is the value of $\cos 2 \alpha$? [3 points]
(1) $- \frac { 1 } { 32 }$
(2) $- \frac { 1 } { 16 }$
(3) $- \frac { 1 } { 8 }$
(4) $- \frac { 1 } { 4 }$
(5) $- \frac { 1 } { 2 }$

[Calculus] When $\tan \theta = - \sqrt { 2 }$, what is the value of $\sin \theta \tan 2 \theta$? (where $\frac { \pi } { 2 } < \theta < \pi$ ) [3 points]
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $\sqrt { 3 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 3 }$
(5) $2 \sqrt { 3 }$

When $\sin \theta = \frac { 1 } { 3 }$, what is the value of $\sin 2 \theta$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [2 points]
(1) $\frac { 7 \sqrt { 2 } } { 18 }$
(2) $\frac { 4 \sqrt { 2 } } { 9 }$
(3) $\frac { \sqrt { 2 } } { 2 }$
(4) $\frac { 5 \sqrt { 2 } } { 9 }$
(5) $\frac { 11 \sqrt { 2 } } { 18 }$

When $\tan \theta = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos 2 \theta$? [2 points]
(1) $\frac { \sqrt { 2 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 5 } } { 3 }$
(5) $\frac { \sqrt { 6 } } { 3 }$

If $\sin \alpha = \frac { 1 } { 3 }$, then $\cos 2 \alpha =$
A. $\frac { 8 } { 9 }$
B. $\frac { 7 } { 9 }$
C. $- \frac { 7 } { 9 }$
D. $- \frac { 8 } { 9 }$

$\cos \mathrm { x } = \frac { - 4 } { 5 }$ Given this, what is $\cos 2 \mathrm { x }$?
A) $\frac { 3 } { 5 }$
B) $\frac { 5 } { 13 }$
C) $\frac { 12 } { 13 }$
D) $\frac { 24 } { 25 }$
E) $\frac { 7 } { 25 }$

$\cos x = \frac { \sqrt { 5 } } { 3 }$
Accordingly, I. $\sin \mathrm { x }$ II. $\sin 2 x$ III. $\cos 2 x$ Which of the following values equals a rational number?
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III