Aula de Coordenadas polares

!"#$%&'()%(*++,)%-')'#(.+/',%#0
!"#$%&'()%(*++,)%-')'#(.+/',%#0
1& !"!#$%& '$ ())*'$+&'&! ,)-&*$! -2& ./'-+

1& !"!#$%& '$ ())*'$+&'&! ,)-&*$! -2& ./'-+
*+-#"#$% %& 2&!"#$%&'()%(*++,)%-')'#(.+/',%#0
.+-$+ O 3"4+5 *6'&')+ )% ,.-) 7+2 )*"/$%8 %
O 3"4+5 *6'&')+ )% ,.-) 7+2 )*"/$%8 %
*+-#"#$%
%&
2&
.+-$+
)% 2& ,'"+ 92% .',$% )+ .:/+5 *6'&')+ )% $"0) ,)-&*;
!"!#$%&
'$ ())*'$+&'&!
-2& ./'-+
)% 2& ,'"+ 1&
92% .',$%
)+ .:/+5
*6'&')+ )%,)-&*$!
$"0) ,)-&*;

..

*+-#"#$% %& 2& .+-$+OO 3"4+5 *6'&')+ )% ,.-) 7+2 )*"/$%8 %
O )+ .:/+5 *6'&')+ )% $"0) ,)-&*;
)% 2& ,'"+ 92% .',$%
<:/+
="4+(.+/',

.

<:/+
="4+(.+/',

>2&

$'/

#"#$%&'O)% *++,)%-')'#5 .+)%&+#

'##+*"', '

.+)%&+#
'##+*"',
*')'>2&
.+-$+$'/P #"#$%&'
-+ ./'-+)%
2&*++,)%-')'#5
.', )%="4+(.+/',
())*'$+&'&!
,)-&*$!
( r ,θ )'5
<:/+

*')'
.+-$+
-+ ./'-+ )%
2&P.',
())*'$+&'&!
,)-&*$!
( r%"4+
,θ )5 r ? P' )"#$@-*"'
+-)%
'+ )%
.:/+
% θ ? + @-A2/+
%-$,% +
>2& $'/ #"#$%&' )% *++,)%-')'#5 .+)%&+# '##+*"', ' r ?%'+)"#$@-*"'
+-)%
'+ .:/+r% ?θ*6'&')+
? + @-A2/+
+ %"4+
.+/',
,'"+ OP)%
; BP-2&%,+
)% %-$,%
())*'$+&'&
*')' .+-$+ P -+ ./'-+ 2& .', )% ())*'$+&'&! ,)-&*$! ( r ,θ )5
P %-92'-$+
*&'"&92% θ r ? & ())*'$+&'&
&+/1-&* 7+2
.+/',
% + )%
,'"+
)% ())*'$+&'&
OP; B -2&%,+ θ ? + @-A2/+ r ? ' )"#$@-*"'
+-)%
)% P '+ .:/+? %*6'&')+
%-$,% + %"4+
2+/1-)
,)-&*8
)% P; 92% θ ? r& ())*'$+&'& &+/1-&* 7+2
P
*&'"&)%
%-92'-$+
.+/', % + ,'"+ OP; B -2&%,+
? *6'&')+
P(r ,θ ) )% ())*'$+&'&
P
2+/1-)
,)-&*8
)%
;
*&'"&- )% P %-92'-$+ 92% θr ? & ())*'$+&'& &+/1-&* 7+2
P ( r ,θ )
2+/1-) ,)-&*8 )% P; O rθ O <:/+ θ r ="4+(.+/',P(r ,θ ) θ="4+(.+/', <:/+O

.
..
<:/+

..
.

="4+(.+/',

As coordenadas polares (r,θ ) estabelecem a posição do ponto P em relação