Circunferência Trigonométrica
Autor: Guilherme Miguel Rosa | Publicado: 2013-01-16T20:00:00-02:00





<!--[if gte mso 9]>        <![endif]-->

<!--[if gte mso 9]>        <![endif]-->

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   UnhideWhenUsed="false" QFormat="true" Name="Book Title"/>         <![endif]--><!--[if gte mso 10]> 
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 <![endif]-->Aula com macetes para facilitar na resolução de exercícios:






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EXERCÍCIOS RESOLVIDOS:

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<![endif]-->
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  Normal
  0
  
  
  21
  
  
  false
  false
  false
  
  PT-BR
  X-NONE
  X-NONE
  
   
   
   
   
   
   
   
   
   
  
  
   
   
   
   
   
   
   
   
   
   
   
  
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1 – Expresse em radianos os seguintes arcos:


&nbsp;a) 60º
&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; b) 315º&nbsp; 




Resolução:

a) π&nbsp;
-&nbsp; 180º

&nbsp;
&nbsp; x&nbsp; - &nbsp; 60º




180ºx
= 60ºπ

x
= 60ºπ

&nbsp;
&nbsp; &nbsp; 180º &nbsp;

x
= π

&nbsp;
&nbsp; &nbsp; 3 &nbsp;



O
arco é π/3 rad.






b) π&nbsp; -&nbsp; 180º

&nbsp; &nbsp; x&nbsp; -&nbsp; 315º




180ºx = 315ºπ&nbsp; 

x = 315ºπ

&nbsp; &nbsp; &nbsp; 180º&nbsp;

x
= 35π

&nbsp;
&nbsp; &nbsp;&nbsp; 20&nbsp;



O
arco é 35π/20 rad.&nbsp;







2
- Encontre a primeira determinação positiva dos seguintes arcos:&nbsp;
&nbsp;
a) 1920º&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; b) -432º


Resolução:

a) Dividimos o arco por 360º, o resto dessa divisão é a primeira determinação positiva do arco.




Na
divisão de 1920º por 360º temos 5 como quociente (5 voltas) e 120º como resto, essa é a primeira determinação positiva.
Obs: não reduzimos ao primeiro quadrante, só fazemos isso se precisar encontrar seno, cosseno ou tangente do ângulo. 




Resposta: 120º

b) Quando o arco é negativo, dividimos da mesma forma para obter o resto, porém o resto é o que arco anda no sentido horário.

Na divisão de -432º por 360º temos quociente 1 e resto -72º. Então o arco anda 72º pra trás do 360º, logo sua primeira DP é de 288º (360º - 72º)

Resposta: 288º







3
- (Unifor-CE) O número real m que satisfaz a sentença:

m + 1 = cos3015º&nbsp; é:

m -
2&nbsp;



a) 4 - 3√2



b) 3√2 -
4



c) 3 - √2



d) 4√2 +
3



e) 3√2 +
4&nbsp;




Resolução:

Dividindo
3015º por 360º obtemos resto 135º, logo:

3015º =
135º



Reduzindo
para o primeiro quadrante:

cos135º =
-cos45º = -√2

&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 &nbsp;



m + 1 = -√2

m -
2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2



2(m + 1)
= -√2(m - 2)

2m + 2 =
-m√2 + 2√2

2m + m√2
= 2√2 - 2

m(2 + √2)
= 2√2 - 2

m = 2√2
- 2

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2 + √2 



Racionalizando:

m = 2√2
- 2&nbsp; . (2 - √2) 

&nbsp;
&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; 2 + √2 &nbsp;&nbsp; (2 - √2)



m = 4√2
- 4 - 4 + 2√2&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
4 - 2



m = 6√2
- 8

&nbsp;
&nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp; 2 &nbsp;



m = 2(3√2
- 4)

&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2 &nbsp; 



m = 3√2 -
4




Gabarito
Letra: B 









4 - (UFSC)
Qual o maior valor numérico que y pode assumir na expressão

y =&nbsp;
37 - 2senx ?

&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 3&nbsp;




Resolução:

Na
expressão, temos senx, então devemos pensar no intervalo de existência
do seno, e escolher para substituí-lo o valor que fará a expressão ter o maior
número no numerador.&nbsp;

Entre
[-1,1], que podemos usar para substituir o senx, respeitando sua
condição de existência, -1 é a melhor opção, pois teremos esse valor
multiplicado por -2, ficando com 37 somado a 2. Logo:

y = 37
- 2(-1)

&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 3&nbsp; 




y = 37
+ 2

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
3

&nbsp; 

y = 39

&nbsp; &nbsp; &nbsp; 3&nbsp; &nbsp; &nbsp;&nbsp;&nbsp;




y = 13


AGORA É A SUA VEZ:
Clique no link abaixo e faça o download da lista de exercícios propostos
Circunferência trigonométrica 
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  <w:LsdException Locked="false" Priority="67" SemiHidden="false"
   UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/>
  <w:LsdException Locked="false" Priority="68" SemiHidden="false"
   UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/>
  <w:LsdException Locked="false" Priority="69" SemiHidden="false"
   UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/>
  <w:LsdException Locked="false" Priority="70" SemiHidden="false"
   UnhideWhenUsed="false" Name="Dark List Accent 6"/>
  <w:LsdException Locked="false" Priority="71" SemiHidden="false"
   UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/>
  <w:LsdException Locked="false" Priority="72" SemiHidden="false"
   UnhideWhenUsed="false" Name="Colorful List Accent 6"/>
  <w:LsdException Locked="false" Priority="73" SemiHidden="false"
   UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/>
  <w:LsdException Locked="false" Priority="19" SemiHidden="false"
   UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/>
  <w:LsdException Locked="false" Priority="21" SemiHidden="false"
   UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/>
  <w:LsdException Locked="false" Priority="31" SemiHidden="false"
   UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/>
  <w:LsdException Locked="false" Priority="32" SemiHidden="false"
   UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/>
  <w:LsdException Locked="false" Priority="33" SemiHidden="false"
   UnhideWhenUsed="false" QFormat="true" Name="Book Title"/>
  
  
 
<![endif]--><!--[if gte mso 10]>

 /* Style Definitions */
 table.MsoNormalTable
 {mso-style-name:"Tabela normal";
 mso-tstyle-rowband-size:0;
 mso-tstyle-colband-size:0;
 mso-style-noshow:yes;
 mso-style-priority:99;
 mso-style-parent:"";
 mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
 mso-para-margin:0cm;
 mso-para-margin-bottom:.0001pt;
 mso-pagination:widow-orphan;
 font-size:10.0pt;
 font-family:"Calibri","sans-serif";
 mso-bidi-font-family:"Times New Roman";}

<![endif]-->