Given $\alpha \in \left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$, and the terminal sides of $\alpha$ and $\beta$ are symmetric about the origin, then the maximum value of $\cos \beta$ is \_\_\_\_.

If the point $(a, 0)$ $(a > 0)$ is a center of symmetry of the graph of the function $y = 2\tan\left(x - \frac{\pi}{3}\right)$, then the minimum value of $a$ is
A. $\frac{\pi}{4}$
B. $\frac{\pi}{2}$
C. $\frac{\pi}{3}$
D. $\frac{4\pi}{3}$

If the point $(a, 0)$ $(a > 0)$ is a center of symmetry of the graph of the function $y = 2\tan\left(x - \frac{\pi}{3}\right)$, then the minimum value of $a$ is
A. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{\pi}{2}$
D. $\frac{4\pi}{3}$

Which of the following is true about $\tan(\sin x)$?
(A) $\tan(\sin x) = 1$ has solutions
(B) $\tan(\sin x) \geq 1$ for some $x$
(C) $\tan(\sin x) < 1$ for all $x$
(D) $\tan(\sin x)$ never attains the value $1$

For $\theta \in (0, \pi/2)$, which of the following is true?
(A) $\cos(\sin\theta) < \cos\theta$
(B) $\cos(\sin\theta) < \sin(\cos\theta)$
(C) $\cos(\sin\theta) > \cos\theta$
(D) $\cos(\sin\theta) > \sin(\cos\theta)$

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

If $\alpha = 3 \sin ^ { - 1 } \left( \frac { 6 } { 11 } \right)$ and $\beta = 3 \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
(A) $\quad \cos \beta > 0$
(B) $\quad \sin \beta < 0$
(C) $\quad \cos ( \alpha + \beta ) > 0$
(D) $\quad \cos \alpha < 0$

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (I), (P), (R)
(B) (II), (Q), (T)
(C) (I), (Q), (U)
(D) (II), (R), (S)

Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (III), (R), (U)
(B) (IV), (P), (R), (S)
(C) (III), (P), (Q), (U)
(D) (IV), (Q), (T)

The value of the limit $$\lim_{x \rightarrow \frac{\pi}{2}} \frac{4\sqrt{2}(\sin 3x + \sin x)}{\left(2\sin 2x \sin\frac{3x}{2} + \cos\frac{5x}{2}\right) - \left(\sqrt{2} + \sqrt{2}\cos 2x + \cos\frac{3x}{2}\right)}$$ is $\_\_\_\_$

Let $f : [ 0,2 ] \rightarrow \mathbb { R }$ be the function defined by
$$f ( x ) = ( 3 - \sin ( 2 \pi x ) ) \sin \left( \pi x - \frac { \pi } { 4 } \right) - \sin \left( 3 \pi x + \frac { \pi } { 4 } \right)$$
If $\alpha , \beta \in [ 0,2 ]$ are such that $\{ x \in [ 0,2 ] : f ( x ) \geq 0 \} = [ \alpha , \beta ]$, then the value of $\beta - \alpha$ is $\_\_\_\_$

The set of all values of $\lambda$ for which the equation $\cos ^ { 2 } 2 x - 2 \sin ^ { 4 } x - 2 \cos ^ { 2 } x = \lambda$ has a solution is: (1) $[ - 2 , - 1 ]$ (2) $\left[ - 2 , - \frac { 3 } { 2 } \right]$ (3) $\left[ - 1 , - \frac { 1 } { 2 } \right]$ (4) $\left[ - \frac { 3 } { 2 } , - 1 \right]$

If the tangent at a point P on the parabola $\mathrm { y } ^ { 2 } = 3 \mathrm { x }$ is parallel to the line $x + 2 y = 1$ and the tangents at the points $Q$ and $R$ on the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 1 } = 1$ are perpendicular to the line $x - y = 2$, then the area of the triangle $P Q R$ is: (1) $\frac { 9 } { \sqrt { 5 } }$ (2) $5 \sqrt { 3 }$ (3) $\frac { 3 } { 2 } \sqrt { 5 }$ (4) $3 \sqrt { 5 }$

(Course 2) On a coordinate plane, consider a circle $C$ with the radius of 1 centered at the origin O. We denote by P and Q the points of intersection of $C$ and the radii which are rotated at angles of $\theta$ and $3\theta$ respectively from the positive section of the $x$ axis, where $0 \leqq \theta \leqq \pi$.
Also, we denote by A the point at which the straight line which is perpendicular to the $x$ axis and passes through point P intersects the $x$ axis, and we denote by B the point at which the straight line which is perpendicular to the $x$ axis and passes through point Q intersects the $x$ axis. Furthermore, we denote the length of line segment AB by $\ell$.
(1) When $\theta = \frac { \pi } { 3 }$, we see that $\ell = \frac { \mathbf { A } } { \mathbf { B } }$.
(2) We are to find the maximum value of $\ell$. When we set $\cos \theta = t$ and express $\ell$ in terms of $t$, we have
$$\ell = \left| \mathbf { C } t ^ { \mathbf { D } } - \mathbf { E } t \right| .$$
Next, when we set $g ( t ) = \mathrm { C } t ^ { \mathrm { D } } - \mathrm { E } t$, we have
$$g ^ { \prime } ( t ) = \mathbf { F } \left( \mathbf { G } t ^ { \mathbf { H } } - 1 \right) .$$
Hence, when
$$\cos \theta = \pm \frac { \sqrt { \mathbf { J } } } { \mathbf { J } }$$
$\ell$ is maximized and its value is $\frac { \mathbf { K } \sqrt { \mathbf { L } } } { \mathbf { M } }$.
(3) For $\mathbf { N } \sim \mathbf{S}$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (9) below.
There are two pairs of points P and Q at which $\ell$ is maximized, and their coordinates are
$$\mathrm { P } \left( \frac { \sqrt { \mathbf{I} } } { \mathbf{J} } , \mathbf{N} \right) \text{ and } \mathrm { Q } \left( \mathbf{O} , \mathbf{P} \right)$$
and
$$\mathrm { P } \left( - \frac { \sqrt { \mathbf{I} } } { \mathbf{J} } , \mathbf{Q} \right) \text{ and } \mathrm { Q } \left( \mathbf{R} , \mathbf{S} \right)$$
(0) $\frac { \sqrt { 6 } } { 3 }$
(1) $\frac { \sqrt { 6 } } { 2 }$
(2) $\frac { 4 \sqrt { 3 } } { 9 }$
(3) $- \frac { 4 \sqrt { 3 } } { 9 }$
(4) $\frac { 5 \sqrt { 3 } } { 9 }$
(5) $- \frac { 5 \sqrt { 3 } } { 9 }$ (6) $\frac { \sqrt { 6 } } { 9 }$ (7) $- \frac { \sqrt { 6 } } { 9 }$ (8) $\frac { 2 \sqrt { 6 } } { 9 }$ (9) $- \frac { 2 \sqrt { 6 } } { 9 }$

On the coordinate plane, a circle with center at the origin $O$ and radius 1 intersects the positive directions of the coordinate axes at points $A$ and $B$ respectively. On the circular arc in the first quadrant, a point $C$ is taken to draw a tangent line to the circle that intersects the two axes at points $D$ and $E$ respectively, as shown in the figure. Let $\angle OEC = \theta$. Select the option that represents $\tan \theta$.
(1) $\overline{OE}$
(2) $\overline{OC}$
(3) $\overline{OD}$
(4) $\overline{CE}$
(5) $\overline{CD}$

Let $f(x) = \sin x + \sqrt{3} \cos x$. Select the correct options.
(1) The vertical line $x = \frac{\pi}{6}$ is an axis of symmetry of the graph of $y = f(x)$
(2) If the vertical lines $x = a$ and $x = b$ are both axes of symmetry of the graph of $y = f(x)$, then $f(a) = f(b)$
(3) In the interval $[0, 2\pi)$, there is only one real number $x$ satisfying $f(x) = \sqrt{3}$
(4) In the interval $[0, 2\pi)$, the sum of all real numbers $x$ satisfying $f(x) = \frac{1}{2}$ does not exceed $2\pi$
(5) The graph of $y = f(x)$ can be obtained from the graph of $y = 4\sin^{2}\frac{x}{2}$ by appropriate (left-right, up-down) translation

Let $\Gamma$ be the function graph of $y = \sin \pi x$ for $0 \leq x \leq 3$. A horizontal line $L : y = k$ intersects $\Gamma$ at three points $P \left( x _ { 1 } , k \right) , Q \left( x _ { 2 } , k \right) , R \left( x _ { 3 } , k \right)$ satisfying $x _ { 1 } < x _ { 2 } < 1 < x _ { 3 }$. Select the correct options.
(1) $k > 0$
(2) $L$ and $\Gamma$ have exactly 3 intersection points
(3) $x _ { 1 } + x _ { 2 } < 1$
(4) If $2 \overline { P Q } = \overline { Q R }$, then $k = \frac { 1 } { 2 }$
(5) The sum of $x$-coordinates of all intersection points of $L$ and $\Gamma$ is greater than 5

On a certain day at a certain location, the duration of daylight (from sunrise to sunset) is exactly 12 hours. The UVI value at that location $x$ hours after sunrise ($0 \leq x \leq 12$) can be expressed by the function $f ( x ) = a \sin ( b x )$ , where $a , b > 0$ . Assume that the UVI value is positive during daylight and 0 during non-daylight hours (i.e., $f ( 0 ) = f ( 12 ) = 0$), and the UVI value 2 hours after sunrise on that day is 4. Find the values of $a$ and $b$.

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $$f(x) = \begin{cases} 2\sin x, & \text{if } \sin x \geq 0 \\ 0, & \text{if } \sin x < 0 \end{cases}$$ Accordingly, which of the following is the image of the open interval $(-\pi, \pi)$ under $f$?
A) $[-2,2]$
B) $(-1,2)$
C) $[0,1]$
D) $(0,2)$
E) $[0,2]$

$$\frac { \cos 135 ^ { \circ } + \cos 330 ^ { \circ } } { \sin 150 ^ { \circ } }$$
What is the value of this expression?
A) $\sqrt { 3 } - \sqrt { 2 }$
B) $\sqrt { 3 } - 1$
C) $\sqrt { 2 } - 1$
D) $\sqrt { 2 } + 1$
E) $\sqrt { 2 } + \sqrt { 3 }$

$\cos x = \frac { \sqrt { 5 } } { 3 }$
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

Let $\mathrm { a } \in \left( \frac { \pi } { 12 } , \frac { \pi } { 6 } \right)$.
$$\begin{aligned} & x = \sin ( 3 a ) \\ & y = \cos ( 3 a ) \\ & z = \tan ( 3 a ) \end{aligned}$$
What is the correct ordering of the numbers?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $y < z < x$
E) $z < x < y$

An isosceles triangle shaped "Beware of Dog!" sign with equal length blue and red edges is hung on a rectangular garden wall with a nail at one corner as shown in the figure.
This sign, which can rotate around the nail, from the position shown in the figure

• if rotated $75^{\circ}$ clockwise, the black edge,
• if rotated $40^{\circ}$ counterclockwise, the blue edge,
• if rotated $x^{\circ}$ clockwise, the red edge

becomes parallel to the top edge of the wall for the first time.
Accordingly, what is x?
A) 5 B) 10 C) 15 D) 20 E) 25

Let $a = \sin(40^{\circ})$
$$\begin{aligned} & b = \sec(40^{\circ}) \\ & c = \tan(40^{\circ}) \end{aligned}$$
Which of the following is the correct ordering of the numbers $a$, $b$ and $c$?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $b < c < a$ E) $c < a < b$