Exercícios sobre equações trigonométricas

Resolver as equações a seguir:

$\,cossec\;x\;=\;cossec\;\dfrac{\,2\pi\,}{\;3\;}\,$

$\,sen\;x\;=\;\dfrac{\,\sqrt{\,3\,}\,}{\;2\;}\,$

$\,sen\,x\;=\;1\,$

$\,sen\,x\;=\;-1\,$

$\,sen\,x\;=\;\dfrac{\;1\;}{\;2\;}\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\;=\;\frac{2\pi}{3}\,+\,2k\pi\;{\text ou }\,x\,=\,\frac{\pi}{3}\,+\,2k\pi \rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\;=\;\frac{\pi}{3}\,+\,2k\pi\;{\text ou }\,x\,=\,\frac{2\pi}{3}\,+\,2k\pi \rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\;=\;\frac{\pi}{2}\,+\,2k\pi \rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\;=\;\frac{3\pi}{2}\,+\,2k\pi\,\rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\;=\;\frac{\pi}{6}\,+\,2k\pi\;{\text ou }\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi \rbrace\,$

×

Resolver em $\,{\rm I\!R}\,$ a equação $\phantom{X}sen\,x\;=\;sen\,\dfrac{\;\pi\;}{\;5\;}\phantom{X}$

resposta:

ciclo trigonométrico senx igual sen pi sobre 5

1. x pode ser:
$\,x\,=\,\dfrac{\,\pi\,}{5}\,+\,2k\pi,\,k\,\in\,\mathbb{Z}\,$ ou
2. x pode ser também:
$\,x\,=\,\left(\pi\,-\,\dfrac{\,\pi\,}{5}\right)\,+\,2k\pi\,=\,$$\dfrac{\,4\pi\,}{5}\,+\,2k\pi,\,k\,\in\,\mathbb{Z}\,$

$\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,\frac{\pi}{5}\,+\,2k\pi\,$ $\,{\text ou}\,x\,=\,\frac{4\pi}{5}\,+\,2k\pi,\phantom{X}k\,\in\,\mathbb{Z}\rbrace\,$
×

Resolver em $\,{\rm I\!R}\,$ a equação $\phantom{X}sen\,x\;=\;\dfrac{\;1\;}{\;2\;}\phantom{X}$

resposta:

Devemos notar que $\,\dfrac{\,1\,}{\,2\,}\,=\,sen\,\dfrac{\,\pi\,}{\,6}\,$, então a equação torna-se $\phantom{X}sen\,x\;=\;\,sen\,\dfrac{\,\pi\,}{\,6}\phantom{X}$
$\,\left\{\begin{array}{rcr} x\,= & \dfrac{\,\pi\,}{\,6\,}\,+\,2\,k\pi \phantom{XXXX} \\ ou \\ x\,= & \left(\,\pi\,-\,\dfrac{\,\pi\,}{\,6\,}\,\right)\,+\,2\,k\pi \\ \end{array} \right.\,$
$\,k\,\in\,\mathbb{Z}\,$
Donde obtemos o conjunto solução:

$\,\mathbb{S}\,=\,\lbrace\,x\,\in\,{\rm I\!R}\phantom{X}|\phantom{X}x\,=\,\dfrac{\pi}{6}\,+\,2k\pi\;$ ou $\;x\,=\,\dfrac{5\pi}{6}\,+\,2k\pi,\,k\,\in\,\mathbb{Z}\rbrace\,$
×

Resolver em $\,{\rm I\!R}\,$ a equação $\phantom{X}cos\,x\;=\;-\,\dfrac{\;\sqrt{\,3\,}\;}{\;2\;}\phantom{X}$

resposta:

Devemos notar que $\,-\,\dfrac{\,\sqrt{\,3\,}\,}{\,2\,}\,=\,cos\,\dfrac{\,5\pi\,}{\,6}\,$, então a equação torna-se $\phantom{X}cos\,x\;=\;\,cos\,\dfrac{\,5\pi\,}{\,6}\phantom{X}$
$\,\left\{\begin{array}{rcr} x\,= \pm\,\dfrac{\,5\pi\,}{\,6\,}\,+\,2\,k\pi \\ \,k\,\in\,\mathbb{Z}\phantom{XXXX} \\ \end{array} \right.\,$
Donde obtemos o conjunto solução:

$\,\mathbb{S}\,=\,\lbrace\,x\,\in\,{\rm I\!R}\phantom{X}|\phantom{X}x\,=\,\pm\,\dfrac{5\pi}{6}\,+\,2k\pi\,,\phantom{X} k\,\in\,\mathbb{Z}\rbrace\,$
×

Resolver em $\,{\rm I\!R}\,$ a equação $\phantom{X}cos\,2x\;=\;0\phantom{X}$

resposta:

Devemos notar que se o cosseno de 2x é zero, então $\,2x = \pm\,\dfrac{\,\pi\,}{\,2\,}\,+\,2k\pi\;\Rightarrow$ $\;x = \pm\,\dfrac{\,\pi\,}{\,4\,}\,\,+\,k\pi,\;k\,\in\,\mathbb{Z}\,$
O conjunto solução então:

$\,\mathbb{S}\,=\,\lbrace\,x\,\in\,{\rm I\!R}\phantom{X}|\phantom{X}x\,=\,\pm\,\dfrac{\pi}{4}\,+\,k\pi\,,\phantom{X} k\,\in\,\mathbb{Z}\rbrace\,$
×

Resolver em $\,{\rm I\!R}\,$ a equação $\phantom{X}tg\,2x\;=\;1\phantom{X}$

resposta:

Devemos notar que se a tangente de 2x é 1, então $\,tg 2x = tg\,\dfrac{\,\pi\,}{\,4\,}\,$
Temos então:
$\,2x\,=\,\dfrac{\,\pi\,}{\,4\,}\,+\,k\pi\;\Rightarrow$ $\;x = \dfrac{\,\pi\,}{\,8\,}\,\,+\,\dfrac{k\pi}{2},\;k\,\in\,\mathbb{Z}\,$
O conjunto solução então:

$\,\mathbb{S}\,=\,\lbrace\,x\,\in\,{\rm I\!R}\phantom{X}|\phantom{X}x\,=\,\dfrac{\pi}{8}\,+\,\dfrac{k\pi}{2}\, ;\phantom{X} k\,\in\,\mathbb{Z}\rbrace\,$
×

Resolver em $\,{\rm I\!R}\,$ as seguintes equações:

$\,sen^2\,x\,=\,\dfrac{\;1\;}{\;4\;}\,$

$\,sen^2\,x\;-\;sen\,x\;=\;0\,$

$\,2\,\centerdot\,sen^2\,x\,-\,3\,\centerdot\,sen\,x\,+\,1\,=\,0\,$

$\,2\,\centerdot\,cos^2\,x\,=\,1\,-\,sen\,x\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{7\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,- \frac{\pi}{6}\,+\,2k\pi\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\,\pi\;$ ou $\,x\,=\,\frac{\pi}{2}\,+\,2k\pi\,\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{2}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\,\rbrace\,$
d)$\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{2}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{-\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{7\pi}{6}\,+\,2k\pi\,\rbrace\,$

×

Resolver as equações:

$\,sen\,x\,=\,sen\,\dfrac{\,\pi\,}{7}\,$

$\,cossec\,x\,=\,2\,$

$\,2\,\centerdot\,sen^2\,x\,=\,1\,$

$\,sen^2\,x\,=\,1\,$

$\,2 \centerdot sen^2\,x\,+\,sen\,x\,-\,1\,=\,0\,$

$\,2 \centerdot sen\,x\,-\,cossec\,x\,=\,1\,$

$\,3 \centerdot tg\,x\,=\,2 \centerdot cos\,x\,$

$\,sen\,x\,+\,cos\,2x\,=\,1\,$

$\,cos^2\,x\,=\,1\,-\,sen\,x\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{7}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{6\pi}{7}\,+\,2k\pi\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\,\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\pm \frac{\pi}{4}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{3\pi}{4}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{4}\,+\,2k\pi\,\rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{2}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{3\pi}{2}\,+\,2k\pi\,\rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{3\pi}{2}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\,\rbrace\,$
f) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{2}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{7\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,2k\pi - \frac{\pi}{6}\,\rbrace\,$
g) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\,\rbrace\,$
h) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\;$ ou $\,x\,=\,\frac{\pi}{6}\,+\,2k\pi\;$ ou $\,x\,=\,\frac{5\pi}{6}\,+\,2k\pi\,\rbrace\,$
i) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\;$ ou $\,x\,=\,\frac{\pi}{2}\,+\,2k\pi\,\rbrace\,$
×

(FEFAAP - 1977) Determinar os valores de x que satisfazem a equação $\phantom{X}4\,sen^{\large\,4}\,x\,-\,11\,sen^{\large\,2}\,x\,+\,6\,=\,0\phantom{X}$

resposta: $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,\pm\,\dfrac{\pi}{3}\,+\,k\pi\,\rbrace\,$
×

Resolver as equações:

$\,sen\,2x\,=\,\dfrac{\,1\,}{\,2\,}\,$

$\,sen\,3x\,=\,\dfrac{\sqrt{\,2\,}}{2}\,$

$\,sen\,(x\,-\,\dfrac{\pi}{3})\,=\,\dfrac{\,\sqrt{\,3\,}\,}{2}\,$

$\,sen\,2x\,=\,sen\,x\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,\frac{\pi}{12}\,+\,k\pi\,$ ou $x\,=\,\frac{5\pi}{12}\,+\,k\pi\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,\frac{\pi}{12}\,+\,\frac{2k\pi}{3}\,$ ou $x\,=\,\frac{\pi}{4}\,+\,\frac{2k\pi}{3}\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,\frac{2\pi}{3}\,+\,2k\pi\,$ ou $x\,=\,\pi\,+\,2k\pi\rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,2k\pi\,$ ou $x\,=\,\frac{\pi}{3}\,+\, \frac{2k\pi}{3}\rbrace\,$
×

Determinar o valor de $\phantom{X}x\;,\;\,x\,\in\,{\rm I\!R}\phantom{X}$ nas seguintes igualdades:

a) $\,sen\,5x\,=\,sen\,3x\phantom{XXXXX}$ b) $\,sen\,3x\,=\,sen\,2x\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,k\pi\,$ ou $x\,=\,\frac{\pi}{8}\,+\,\frac{k\pi}{4} \rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\,|\,x\,=\,2k\pi\,$ ou $x\,=\,\frac{\pi}{5}\,+\,\frac{2k\pi}{5}\rbrace\,$
×

Determinar os ângulos internos de um triângulo sabendo que estão em progressão aritmética e que o seno da soma do menor ângulo com o ângulo médio é $\phantom{X}\dfrac{\,\sqrt{\,3\,}\,}{\,2\,}\phantom{X}$

resposta: ângulos $\,\frac{\pi}{3},\,\frac{\pi}{3},\,\frac{\pi}{3}\,$
×

(MAPOFEI - 1976) Resolver o sistema $\,\left\{\begin{array}{rcr} sen\,(x\,+\,y)\,=\,0 & \\ x\,-\,y\,=\,\pi \phantom{XXX} & \\ \end{array} \right.\,$

resposta: $\,x\,=\,\frac{\pi}{2}\,+\,\frac{k\pi}{2}\,$, $\,y\,=\,-\frac{\pi}{2}\,+\,\frac{k\pi}{2}\,$
×

Resolver as equações:

$\,cos\,x\,=\,cos\,\dfrac{\,\pi\,}{5}\,$

$\,sec\,x\,=\,sec\,\dfrac{\,2\pi\,}{3}\,$

$\,cos\,x\,=\,0\,$

$\,cos\,x\,=\,1\,$

$\,cos\,x\,=\,-1\,$

$\,cos\,x\,=\,\dfrac{\,1\,}{2}\,$

$\,cos\,x\,=\,\dfrac{\,\sqrt{\,2\,}\,}{2}\,$

$\,cos\,x\,=\,\dfrac{\,\sqrt{\,3\,}\,}{2}\,$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{5}\,+\,2k\pi \, \rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{2\pi}{3}\,+\,2k\pi \, \rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{2}\,+\,2k\pi \, \rbrace\, =\,$ $\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{2}\,+\,k\pi \, \rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,2k\pi \, \rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pi\,+\,2k\pi \, \rbrace\,$
f) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{3}\,+\,2k\pi \, \rbrace\,$
g) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{4}\,+\,2k\pi \, \rbrace\,$
h) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{5\pi}{6}\,+\,2k\pi \, \rbrace\,$

×

Resolver as equações:

$\phantom{X}4 \centerdot cos^2 x = 3\phantom{X}$

$\phantom{X}cos^2\,x\,+\,cos\,x\,=\,0\phantom{X}$

$\phantom{X}sen^2\,x\,=\,1\,+\,cos\,x\phantom{X}$

$\phantom{X}cos\,2x\,+\,3\,\centerdot\,cos\,x\,+\,2\,=\,0\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{6}\,+\,2k\pi \, ou \, x\,=\,\pm\,\frac{5\pi}{6}\,+\,2k\pi\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{2}\,+\,k\pi\;ou\;x\,=\,\pi\,+\,2k\pi \, \rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{2}\,+\,k\pi\;ou\;x\,=\,\pi\,+\,2k\pi \, \rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pi + 2k\pi \; ou \; x\,=\,\pm\,\frac{2\pi}{3}\,+\,2k\pi \rbrace\,$

×

Resolver as seguintes equações:

$\phantom{X}cos x = -\dfrac{ 1 }{ 2 }\phantom{X}$

$\phantom{X}cos\,x\,=\,-\,\dfrac{\,\sqrt{\,2\,}\,}{\,2\,}\phantom{X}$

$\phantom{X}cos\,x\,=\,\dfrac{\,\sqrt{\,3\,}\,}{\,2\,}\phantom{X}$

$\phantom{X}sec\,x\,=\,\,2\phantom{X}$

$\phantom{X}2\,\centerdot\,cos^2\,x\,=\,cos\,x\phantom{X}$

$\phantom{X}4\,\centerdot\,cos\,x\,+\,3\,sec\,x\,=\,8\phantom{X}$

$\phantom{X}2\,-\,2\,\centerdot\,cos\,x\,=\,sen\,x\,\centerdot\,tg\,x\phantom{X}$

$\phantom{X}2\,\centerdot\,sen^2\,x\,+\,6\,\centerdot\,cos\,x\,=\,5\,+\,cos\,2x\phantom{X}$

$\phantom{X}1\,+\,3\,\centerdot\,tg^2\,x\,=\,5\,\centerdot\,sec\,x\phantom{X}$

$\phantom{X}\left(\,4\,-\,\dfrac{3}{sen^2x}\right)\,\centerdot\,\left(\,4\,-\,\dfrac{1}{cos^2x}\,\right)\,=\,0\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{2\pi}{3}\,+\,2k\pi\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{3\pi}{4}\,+\,2k\pi\, \rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{6}\,+\,2k\pi \rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi \rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi\,ou\,x\,=\,\frac{\pi}{2}\,+\,k\pi\,\rbrace\,$
f) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi \rbrace\,$
g) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,2k\pi \rbrace\,$
h) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,2k\pi\,ou\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi \rbrace\,$
i) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi \rbrace\,$
j) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\frac{\pi}{3}\,+\,2k\pi \; ou \; x\,=\,\pm\frac{2\pi}{3}\,+\,2k\pi\rbrace\,$

×

Resolver as equações a seguir:

$\phantom{X}cos\,2x\,=\,\dfrac{\,\sqrt{\,3\,}\,}{\,2\,}\phantom{X}$

$\phantom{X}cos\,2x\,=\,cos\,x\phantom{X}$

$\phantom{X}cos\left(x\,+\,\dfrac{\,\pi\,}{\,6\,}\right) = 0\phantom{X}$

$\phantom{X}cos\left(x\,-\,\dfrac{\,\pi\,}{\,4\,}\right)\,=\,1\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\pm\,\frac{\pi}{12}\,+\,k\pi\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,2k\pi \; ou\; x\,=\frac{2k\pi}{3}\, \rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{3}\,+\,2k\pi\;ou\;,x\,=\,-\frac{2\pi}{3}\,+\,2k\pi \rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{4}\,+\,2k\pi \rbrace\,$

×

Resolver as equações:

$\phantom{X}cos\,3x\,-\,cos\,x\,=\,0\phantom{X}$

$\phantom{X}cos\,5x\,=\,cos\left(x\,+\,\dfrac{\,\pi\,}{\,3\,}\right)\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,k\pi\;ou\;x\,=\,\,\frac{k\pi}{2}\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm\,I\!R}\;|\;\,x\,=\,\frac{\pi}{12} + \frac{k\pi}{2} \; ou\; x\,=\frac{-\pi}{18}\,+\,\frac{k\pi}{3} \rbrace\,$

×

Determinar os ângulos internos de um triângulo ABC sabendo que $\phantom{X}cos(A\,+\,B)\,=\,\dfrac{\,1\,}{\,2\,}\phantom{X}$ e $\phantom{X}sen(B\,+\,C)\,=\,\dfrac{\,1\,}{\,2\,}\phantom{X}$

resposta: A = π/6, B = π/6 e C = 2π/3
×

(MAUÁ - 1977) Dada a equação $\phantom{X}(sen\,x\,+\,cos\,y)(sec\,x\,+\,cossec\,y)\,=\,4\phantom{X}$:

a) resolva-a se $\phantom{X}x\,=\,y\phantom{X}$
b) resolva-a se $\phantom{X}sen\,x\,=\,cos\,y\phantom{X}$

resposta: a) x = y = π/4 + kπ b) x = π/4 + kπ e y + x = π/2 + 2kπ
×

Resolver as equações:

$\phantom{X}tg\,x\,=\,1\phantom{X}$

$\phantom{X}cotg\,x\,=\,\sqrt{\,3\,}\phantom{X}$

$\phantom{X}tg\,x\,=\,-\sqrt{\,3\,}\phantom{X}$

$\phantom{X}tg\,x\,=\,0\phantom{X}$

$\phantom{X}tg\,2x\,=\,\sqrt{\,3\,}\phantom{X}$

$\phantom{X}tg\,2x\,=\,tg\,x\phantom{X}$

$\phantom{X}tg\,3x\,=\,1\phantom{X}$

$\phantom{X}tg\,5x\,=\,tg\,3x\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{4}\,+\,k\pi\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,k\pi\,\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{2\pi}{3}\,+\,k\pi\,\rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\,\rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,\frac{k\pi}{2}\,\rbrace\,$
f) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\,\rbrace\;$
g) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{12}\,+\,\frac{k\pi}{3}\,\rbrace\,$
h) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{k\pi}{2}, k é par\,\rbrace\,$

×

Resolver as equações:

$\phantom{X}sen\,x\,\,-\,\sqrt{\,3\,}\,\centerdot\,cosx\,=\,0\phantom{X}$

$\phantom{X}sen^2\,x\,=\,cos^2\,x\phantom{X}$

$\phantom{X}tg\,x\,+\,cotg\,x\,=\,2\phantom{X}$

$\phantom{X}sec^2\,x\,=\,1\;+\;tg\,x\,\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{3}\,+\,k\pi\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{4}\,+\,k\pi\;ou\;x\,=\,\frac{3\pi}{4}\,+\,k\pi;\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{4}\,+\,k\pi\,\rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\;ou\;x\,=\,\frac{\pi}{4}\,+\,k\pi\;\rbrace\,$

×

Resolver as equações:

$\phantom{X}tg\,x\,=\,tg\,\dfrac{\,\pi\,}{5}\phantom{X}$

$\phantom{X}cotg\,x\,=\,cotg\,\dfrac{\,5\pi\,}{6}\phantom{X}$

$\phantom{X}3\,\centerdot\,tg\,x\,=\,\sqrt{\,3\,}\phantom{X}$

$\phantom{X}cotg\,x\,=\,0\phantom{X}$

$\phantom{X}cotg\,x\,=\,-1\phantom{X}$

$\phantom{X}tg\,3x\,-\,tg\,2x\,=\,0\phantom{X}$

$\phantom{X}tg\,2x\,=\,tg\,(x\,+\,\dfrac{\pi}{4})\phantom{X}$

$\phantom{X}tg\,4x\,=\,1\phantom{X}$

$\phantom{X}cotg\,2x\,=\,cotg(x\,+\,\dfrac{\pi}{4})\phantom{X}$

$\phantom{X}tg^2\,2x\,=\,3\phantom{X}$

resposta: a) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{5}\,+\,k\pi\,\rbrace\,$
b) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{5\pi}{6}\,+\,k\pi\;\rbrace\,$
c) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,k\pi\;\rbrace\,$
d) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{2}\,+\,k\pi\;\rbrace\,$
e) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{3\pi}{4}\,+\,k\pi\;\rbrace\,$
f) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,k\pi\;\rbrace\,$
g) $\,S\,=\,\varnothing\; \Leftarrow$
h) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{16}\,+\,\frac{k\pi}{4}\;\rbrace\,$
i) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{4}\,+\,k\pi\;\rbrace\,$
j) $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\frac{\pi}{6}\,+\,\dfrac{k\pi}{2}\;$ ou $\phantom{X}x = \frac{\pi}{3} + \frac{k\pi}{2}\rbrace\,$

×

Resolver as equações:

$\phantom{X}sec^2\,x\,=\,2\,\centerdot\,tg\,x\phantom{X}$

$\phantom{X}\dfrac{1}{sen^2\,x}\,=\,1\,-\,\dfrac{cos\,x}{sen\,x}\phantom{X}$

$\phantom{X}sen\,2x\,\centerdot\,cos(x\,+\,\dfrac{\pi}{\,4\,})\,=\,cox\,2x\,\centerdot\,sen(x\,+\,\dfrac{\,\pi\,}{4})\phantom{X}$

$\phantom{X}(1\,-\,tg\,x)(1\,+\,sen\,2x)\,=\,1\,+\,tg\,x\phantom{X}$

(MAPOFEI - 1975) Resolver a equação $\phantom{X}cotg\;x\;-\;sen\;2x\;=\;0\phantom{X}$.

resposta: $\,S\,=\,\lbrace\,x\,\in\,{\rm I\!R}\;|\;x\,=\,\pm\frac{\pi}{4}\,+\,k\pi\,\rbrace\,$

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(FEI - 1977) Para quais valores de $\,p\,$ a equação $\phantom{X}tg\;px\,=\,cotg\;px\phantom{X}$ tem $\,x\,=\,\dfrac{\,\pi\,}{\,2\,}\,$ para raiz?

resposta: $\,p = \frac{1}{2}\,+\,k,\,k\,\in\,\mathbb{Z}\,$

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