Question 180
A figura mostra o gráfico da função seno no intervalo $[0, 2\pi]$.
[Figure]
O número de soluções da equação $\operatorname{sen}(x) = 0{,}5$ no intervalo $[0, 2\pi]$ é
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Find all $x \in [ - \pi , \pi ]$ such that $\cos 3 x + \cos x = 0$.

Given $\cos x + \cos y + \cos z = \frac { 3 \sqrt { } 3 } { 2 }$ and $\sin x + \sin y + \sin z = \frac { 3 } { 2 }$ then show that $x = \frac { \pi } { 6 } + 2 k \pi , y = \frac { \pi } { 6 } + 2 \ell \pi , z = \frac { \pi } { 6 } + 2 m \pi$ for some $k , \ell , m \in \mathbf { Z }$.

The number of $\theta$ with $0 \leq \theta < 2 \pi$ such that $4 \sin ( 3 \theta + 2 ) = 1$ is
(A) 2
(B) 3
(C) 6
(D) none of the above

(Calculus) For $0 \leqq x < 2\pi$, the sum of all distinct values of $x$ satisfying the equation $\sin 2x = 2\cos x - 2\cos^2 x$ is? [3 points]
(1) $\pi$
(2) $\frac{5}{4}\pi$
(3) $\frac{3}{2}\pi$
(4) $\frac{7}{4}\pi$
(5) $2\pi$

When $0 < x < 2 \pi$, the sum of all real roots of the equation $\cos ^ { 2 } x - \sin x = 1$ is $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [3 points]

When $0 \leq x < 2 \pi$, what is the sum of all solutions to the equation $$\cos ^ { 2 } x = \sin ^ { 2 } x - \sin x$$ ? [3 points]
(1) $2 \pi$
(2) $\frac { 5 } { 2 } \pi$
(3) $3 \pi$
(4) $\frac { 7 } { 2 } \pi$
(5) $4 \pi$

For $0 < x < 2 \pi$, what is the sum of all values of $x$ that simultaneously satisfy the equation $4 \cos ^ { 2 } x - 1 = 0$ and the inequality $\sin x \cos x < 0$? [3 points]
(1) $\frac { 10 } { 3 } \pi$
(2) $3 \pi$
(3) $\frac { 8 } { 3 } \pi$
(4) $\frac { 7 } { 3 } \pi$
(5) $2 \pi$

For $0 \leq x < 4 \pi$, what is the sum of all solutions to the equation $$4 \sin ^ { 2 } x - 4 \cos \left( \frac { \pi } { 2 } + x \right) - 3 = 0$$ ? [4 points]
(1) $5 \pi$
(2) $6 \pi$
(3) $7 \pi$
(4) $8 \pi$
(5) $9 \pi$

4. Use a 2B pencil to answer multiple-choice questions, and use a black pen, marker, or ballpoint pen to answer non-multiple-choice questions.
I. Fill-in-the-Blank Questions (Total Score: 48 points, 4 points each)
1. If $\operatorname { tg } \alpha = \frac { 1 } { 2 }$, then $\operatorname { tg } \left( \alpha + \frac { \pi } { 4 } \right) =$ $\_\_\_\_$.
2. A parabola has vertex at $(2,0)$ and directrix $x = -1$. Its focus is at $\_\_\_\_$.
3. Let $A = \left\{ 5 , \log _ { 2 } ( a + 3 ) \right\}$ and $B = \{ a , b \}$. If $A \cap B = \{ 2 \}$, then $A \cup B =$ $\_\_\_\_$.
4. For a geometric sequence $\left\{ a _ { n } \right\} ( n \in \mathbb{N} )$ with common ratio $q = - \frac { 1 } { 2 }$, if $\lim _ { n \rightarrow \infty } \left( a _ { 1 } + a _ { 3 } + a _ { 5 } + \cdots + a _ { 2 n - 1 } \right) = \frac { 8 } { 3 }$, then $a _ { 1 } =$ $\_\_\_\_$.

14. Given that $y = f ( x )$ is a function with period $2 \pi$, and when $x \in [ 0,2 \pi )$, $f ( x ) = \sin \frac { x } { 2 }$, the solution set of $f ( x ) = \frac { 1 } { 2 }$ is
A. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
B. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { 5 \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
C. $\left\{ x \left\lvert \, x = 2 k \pi \pm \frac { \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
D. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { \pi } { 3 } + ( - 1 ) ^ { k } \frac{\pi}{3} \right. , k \in \mathbb{Z} \right\}$.

17. If the solution sets of the trigonometric equations $\sin x = 0$ and $\sin 2x = 0$ are $E$ and $F$ respectively, then ( )
(A) $E \subset F$
(B) $E \supset F$
(C) $E = F$
(D) $E \cap F = \varnothing$

Let the function $f(x) = \cos(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) have minimum positive period $T$. If $f(T) = \frac{\sqrt{3}}{2}$ and $x = \frac{\pi}{6}$ is a zero of $f(x)$, then the minimum value of $\omega$ 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. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{\pi}{2}$
D. $\frac{4\pi}{3}$

109- The general solution of the trigonometric equation $2\sqrt{2}\sin x \cos x = \sin x + \cos x$ is which of the following?

• [(1)] $k\pi + \dfrac{\pi}{4}$
• [(2)] $\dfrac{2k\pi}{3} - \dfrac{\pi}{4}$
• [(3)] $\dfrac{2k\pi}{3} + \dfrac{\pi}{4}$
• [(4)] $2k\pi \pm \dfrac{\pi}{4}$

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109- What is the general solution of the trigonometric equation $\dfrac{\sin 3x}{\sin x}=2\cos^2 x$?
(1) $\dfrac{k\pi}{2}$ (2) $\dfrac{k\pi}{2}+\dfrac{\pi}{4}$ (3) $k\pi-\dfrac{\pi}{4}$ (4) $k\pi+\dfrac{\pi}{4}$

109- What is the general solution of the trigonometric equation $\cot x = \dfrac{\sin x + \sin 2x}{\cos x + \cos 2x}$?

p{6cm}} (2) $\dfrac{2k\pi}{5}$	(1) $\dfrac{k\pi}{5}$
[18pt] (4) $\dfrac{1}{5}(2k+1)\pi$	(3) $\dfrac{3k\pi}{5}$

108. The sum of all solutions of the equation $\sin 4x = \sin^2 x - \cos^2 x$, in the interval $[0, \pi]$, equals which of the following?
(4) $\dfrac{11\pi}{2}$ (3) $\dfrac{5\pi}{2}$ (2) $\dfrac{9\pi}{4}$ (1) $\dfrac{7\pi}{4}$

110- The general solution of the trigonometric equation $\cos 2x = \sin x \sin 3x$ is which of the following?
(1) $\dfrac{k\pi}{2} - \dfrac{\pi}{6}$ (2) $\dfrac{k\pi}{3} + \dfrac{\pi}{6}$ (3) $k\pi + \dfrac{\pi}{2}$ (4) $\dfrac{k\pi}{3}$

113. What is the solution set of the trigonometric equation $\sin^2 x + \cos^2 x = 1 - \dfrac{1}{2}\sin 2x$ on the interval $[0, 2\pi]$?
(1) $\dfrac{5\pi}{2}$ (2) $\dfrac{7\pi}{2}$ (3) $2\pi$ (4) $3\pi$

112. The sum of the solutions of the equation $\tan(3x)\tan(x) = 1$ in the interval $[\pi, 2\pi]$ is:
$$5\pi \quad (1) \qquad 6\pi \quad (2) \qquad \frac{9\pi}{2} \quad (3) \qquad \frac{11\pi}{2} \quad (4)$$

107- Suppose $A$ is the solution set of the trigonometric equation $\left(1+\cos(2\alpha)\right)\!\left(1+\cos(4\alpha)\right)\!\left(1+\cos(8\alpha)\right) = \dfrac{1}{8}$, in the interval $[0, \pi]$. What is the maximum element of $A$?
(1) $\dfrac{5}{7}\pi$ (2) $\dfrac{6}{7}\pi$ (3) $\dfrac{7}{9}\pi$ (4) $\dfrac{8}{9}\pi$

In the triangle $ABC$, the angle $\angle BAC$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $ABC$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

The set of all solutions of the equation $\cos 2\theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n\pi + \frac{\pi}{2}$, where $n$ is any integer
(C) $\theta = 2n\pi$ or $\theta = 2n\pi - \frac{\pi}{2}$ or $\theta = n\pi - \frac{\pi}{4}$, where $n$ is any integer
(D) $\theta = 2n\pi$ or $\theta = n\pi + \frac{\pi}{4}$, where $n$ is any integer

The number of solutions of the equation $\sin^{-1} x = 2 \tan^{-1} x$ is
(A) 1
(B) 2
(C) 3
(D) 5