For $x$ satisfying the equation $3 \cos 2 x + 17 \cos x = 0$, find the value of $\tan ^ { 2 } x$. [3 points]

4. If $0 < \theta < 2 \pi$, then the intervals of values of $\theta$ for which $2 \sin ^ { 2 } \theta - 5 \sin \theta + 2 > 0$, is
(A) $\left( 0 , \frac { \pi } { 6 } \right) \cup \left( \frac { 5 \pi } { 6 } , 2 \pi \right)$
(B) $\left( \frac { \pi } { 8 } , \frac { 5 \pi } { 6 } \right)$
(C) $\left( 0 , \frac { \pi } { 8 } \right) \cup \left( \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right)$
(D) $\left( \frac { 41 \pi } { 48 } , \pi \right)$
Sol. (A) $2 \sin ^ { 2 } \theta - 5 \sin \theta + 2 > 0$ $\Rightarrow \quad ( \sin \theta - 2 ) ( 2 \sin \theta - 1 ) > 0$ $\Rightarrow \sin \theta < \frac { 1 } { 2 }$ $\Rightarrow \quad \theta \in \left( 0 , \frac { \pi } { 6 } \right) \cup \left( \frac { 5 \pi } { 6 } , 2 \pi \right)$.

The sum of all values of $\theta \in [ 0,2 \pi ]$ satisfying $2 \sin ^ { 2 } \theta = \cos 2 \theta$ and $2 \cos ^ { 2 } \theta = 3 \sin \theta$ is
(1) $4 \pi$
(2) $\frac { 5 \pi } { 6 }$
(3) $\pi$
(4) $\frac { \pi } { 2 }$

Find the maximum angle $x$ in the range $0 ^ { \circ } \leq x \leq 360 ^ { \circ }$ which satisfies the equation
$$\cos ^ { 2 } ( 2 x ) + \sqrt { 3 } \sin ( 2 x ) - \frac { 7 } { 4 } = 0$$
A $30 ^ { \circ }$ B $60 ^ { \circ }$ C $120 ^ { \circ }$ D $150 ^ { \circ }$ E $210 ^ { \circ }$ F $240 ^ { \circ }$ G $300 ^ { \circ }$ H $330 ^ { \circ }$

It is given that
$$y = ( 1 + 2 \cos x ) \cos 2 x \quad \text { for } 0 < x < \pi$$
The complete set of values of $x$ for which $y$ is negative is
A $0 < x < \frac { \pi } { 4 } , \frac { 2 \pi } { 3 } < x < \frac { 3 \pi } { 4 }$ B $0 < x < \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } < x < \pi$ C $0 < x < \frac { 2 \pi } { 3 } , \frac { 3 \pi } { 4 } < x < \pi$ D $\frac { \pi } { 4 } < x < \frac { 2 \pi } { 3 } , \frac { 3 \pi } { 4 } < x < \pi$ E $\frac { \pi } { 4 } < x < \frac { 2 \pi } { 3 }$ F $\frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$

Find the complete set of values of $x$, with $0 \leq x \leq \pi$, for which
$$( 1 - 2 \sin x ) \cos x \geq 0$$
A $0 \leq x \leq \frac { \pi } { 6 } , \frac { \pi } { 2 } \leq x \leq \frac { 5 \pi } { 6 }$
B $0 \leq x \leq \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \leq x \leq \pi$
C $\frac { \pi } { 6 } \leq x \leq \frac { \pi } { 2 } , \quad \frac { 5 \pi } { 6 } \leq x \leq \pi$
D $\frac { \pi } { 6 } \leq x \leq \frac { 5 \pi } { 6 }$

Find the number of solutions and the sum of the solutions of the equation
$$1 - 2\cos^2 x = |\cos x|$$
where $0 \leq x \leq 180^{\circ}$
A Number of solutions $= 2$ Sum of solutions $= 180^{\circ}$
B Number of solutions $= 2$ Sum of solutions $= 240^{\circ}$
C Number of solutions $= 3$ Sum of solutions $= 180^{\circ}$
D Number of solutions $= 3$ Sum of solutions $= 360^{\circ}$
E Number of solutions $= 4$ Sum of solutions $= 240^{\circ}$
F Number of solutions $= 4$ Sum of solutions $= 360^{\circ}$

The solutions to $7 x ^ { 4 } - 6 x ^ { 2 } + 1 = 0$ are $\pm \cos \theta$ and $\pm \cos \beta$.
Which one of the following equations has solutions $\pm \sin \theta$ and $\pm \sin \beta$ ?