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 $\cos 3 x + \cos x = 0$.

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

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]

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Prove that
$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$

The number of solutions of the equation $\sin ( \cos \theta ) = \theta$, $- 1 \leq \theta \leq 1$, is
(a) 0
(b) 1
(c) 2
(d) 3

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 roots of the equation $x^2 + \sin^2 x = 1$ in the closed interval $\left[ 0, \frac{\pi}{2} \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

Find the number of solutions of $\sec x + \tan x = 2\cos x$ in $[0, 2\pi]$.
(A) 0 (B) 1 (C) 2 (D) 3

The number of solutions of the equation $\sin ( 7 x ) + \sin ( 3 x ) = 0$ with $0 \leq x \leq 2 \pi$ is
(A) 9
(B) 12
(C) 15
(D) 18.

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then
(A) $S$ is the empty set.
(B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$.
(C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$.
(D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.

For $x \in (0, \pi)$, the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution

The number of distinct solutions of the equation $$\frac { 5 } { 4 } \cos ^ { 2 } 2 x + \cos ^ { 4 } x + \sin ^ { 4 } x + \cos ^ { 6 } x + \sin ^ { 6 } x = 2$$ in the interval $[ 0,2 \pi ]$ is

$$\lim_{x\rightarrow 2}\left(\frac{\sqrt{1-\cos\{2(x-2)\}}}{x-2}\right)$$
(1) equals $\sqrt{2}$
(2) equals $-\sqrt{2}$
(3) equals $\frac{1}{\sqrt{2}}$
(4) does not exist

The number of values of $\alpha$ in $[ 0,2 \pi ]$ for which $2 \sin ^ { 3 } \alpha - 7 \sin ^ { 2 } \alpha + 7 \sin \alpha = 2$, is:
(1) 3
(2) 1
(3) 6
(4) 4

If $0 \leq x < 2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is: (1) 3 (2) 5 (3) 7 (4) 9

If sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac { \pi } { 6 } + x \right) \cdot \cos \left( \frac { \pi } { 6 } - x \right) - \frac { 1 } { 2 } \right) = 1$ in $[ 0 , \pi ]$ is $k \pi$, then $k$ is equal to:
(1) $\frac { 20 } { 9 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 13 } { 9 }$
(4) $\frac { 8 } { 9 }$

If $0 \leq x < \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$, is:
(1) 4
(2) 3
(3) 2
(4) 1

Let $S = \left\{ \theta \in [ - 2 \pi , 2 \pi ] : 2 \cos ^ { 2 } \theta + 3 \sin \theta = 0 \right\}$. Then the sum of the elements of $S$ is:
(1) $\pi$
(2) $\frac { 13 \pi } { 6 }$
(3) $\frac { 5 \pi } { 3 }$
(4) $2 \pi$

The number of solutions of $\sin ^ { 7 } x + \cos ^ { 7 } x = 1 , x \in [ 0,4 \pi ]$ is equal to
(1) 11
(2) 7
(3) 5
(4) 9

The number of roots of the equation, $( 81 ) ^ { \sin ^ { 2 } x } + ( 81 ) ^ { \cos ^ { 2 } x } = 30$ in the interval $[ 0 , \pi ]$ is equal to :
(1) 3
(2) 4
(3) 8
(4) 2

The number of solutions of the equation $x + 2 \tan x = \frac { \pi } { 2 }$ in the interval $[ 0,2 \pi ]$ is
(1) 3
(2) 4
(3) 2
(4) 5

Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^ { 4 } \theta + \cos ^ { 4 } \theta - \sin \theta \cos \theta = 0$ in $[ 0,4 \pi ]$ then $\frac { 8 S } { \pi }$ is equal to

The number of solutions of the equation $\cos \left( x + \frac { \pi } { 3 } \right) \cos \left( \frac { \pi } { 3 } - x \right) = \frac { 1 } { 4 } \cos ^ { 2 } 2 x , x \in [ - 3 \pi , 3 \pi ]$ is:
(1) 8
(2) 5
(3) 6
(4) 7