Cap19.pdf

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O ⇢ ✓
O
OA O
P
⇢ ✓ ⇢
P O ✓
OP
P = ( ⇢ , ✓ )
OA OP
(⇢, ✓) = (�⇢, ✓ + ⇡)
⇢
⇢
|⇢|
P = (⇢, ✓) ⇢ < 0
P = (�⇢, ✓ + ⇡)
P = (2, 30o) = (2, ⇡/6).
(⇢, ✓)
✓ ✓ + 2⇡k k 2 Z
(⇢, ✓) (⇢, ✓ + 2⇡k)
(⇢, ✓) = (�⇢, ✓ + ⇡) ⇢ < 0
(�⇢, ✓+ ⇡) = (⇢, ✓+2⇡) = (⇢, ✓) ⇢ > 0 (⇢, ✓) = (�⇢, ✓+ ⇡)
⇢ 2 R
(⇢, ✓) = (�⇢, ✓ + (2k + 1)⇡) k 2 Z ⇢ 2 R
O⇢✓
O⇢✓
P1 = (1, 0o)
P1 O⇢✓
P1 P1 = (�1, 180o) =
(1, 360o k), k 2 Z ⇤
P2 = (4,�⇡/4)
P2 O⇢✓
P2 = (�4,�⇡/4+⇡) = (4,�⇡/4+2⇡k) , k 2 Z
P2 ⇤
P3 = (�1, 0o)
P3 O⇢✓
⇢ = �1 P3 = (1, 0o+180o) = (1, 180o) = (1, ⇡) =
(1, ⇡ + 2⇡k) k 2 Z ⇤
P4 = (�2, ⇡/3)
P4 O⇢✓
⇢ < 0 P4 = (2, ⇡/3 + ⇡) = (2, 4⇡/3 + 2⇡k) , k 2 Z ⇤
O⇢✓
P = (⇢, ✓) ⇢ = 3
⇢ = 3.
⇢
✓
O 3
O 3
⇤
⇢ = �3
⇢ = a
|a|
O⇢✓
r P = (⇢, ✓) ✓ =
⇡
4
r = {(⇢, ✓) | ✓ = ⇡
4
⇢ 2 R}
r O ✓0 =
⇡
4
OA
P1, . . . , P4 r
⇤
O ✓ = ✓0
✓0 ✓ = ✓0 + 2⇡k k 2 Z
O⇢✓
OXY
OX �OY �OX 90o
O⇢✓
OXY
P 6= O
⇢ ✓ O⇢✓
x y OXY
x = ⇢ cos ✓ y = ⇢ sen ✓
x2 = ⇢2 cos2 ✓ , y2 = ⇢2 sen2 ✓ , cos ✓ =
x
⇢
, sen ✓ =
y
⇢
y
x
=
sen ✓
cos ✓
= tg ✓ ,
⇢ =
p
x2 + y2 , cos ✓ =
xp
x2 + y2
, sen ✓ =
yp
x2 + y2
tg ✓ =
y
x
x2 + y2 = ⇢2(cos2 ✓ + sen2 ✓) = ⇢2 ,
⇢ = |⇢| =px2 + y2 ⇢ � 0
⇢ =
p
x2 + y2 cos ✓ =
x
⇢
sen ✓ =
y
⇢
⇢ < 0
⇢0 = �px2 + y2
✓0 cos ✓0 = � xp
x2 + y2
sen ✓0 = � yp
x2 + y2
x = ⇢0 cos ✓0
y = ⇢0 sen ✓0
cos ✓0 = � cos ✓ sen ✓0 = � sen ✓ ✓0 = ✓ + ⇡
(⇢, ✓) (�⇢, ✓ + ⇡)
O⇢✓ OXY
OX
P = (2,⇡/2)
P = (0, 2)
P = (⇢, ✓) = (2, ⇡/2)
⇢ = 2 ✓ = ⇡/2
x = ⇢ cos ✓ = 2 cos ⇡/2 = 0
y = ⇢ sen ✓ = 2 sen ⇡/2 = 2
P
⇤
P = (x, y) = (1, 1)
P = (1, 1)
P = (
p
2,⇡/4)
x = 1 y = 1
⇢ =
p
x2 + y2 =
p
12 + 12 =
p
2
cos ✓ =
1p
2
sen ✓ =
1p
2
✓ = ⇡/4
✓ = ⇡/4 + 2⇡k k 2 Z
P = (⇢, ✓) = (
p
2, ⇡/4) = (
p
2, ⇡/4 + 2⇡k)
P
(�p2, ⇡/4 + (2k + 1)⇡) k 2 Z
P
⇤
P = (�3,⇡/2)
P = (0,�3)
P = (⇢, ✓) = (�3, ⇡/2)
P = (�3, ⇡/2) = (3, ⇡/2+⇡) = (3, 3⇡/2)
x = ⇢ cos ✓ = �3 cos ⇡
2
= 3 cos
3⇡
2
= 0
y = ⇢ sen ✓ = �3 sen ⇡
2
= 3 sen
3⇡
2
= �3
P
⇤
P = (�p2, 5⇡/4)
P = (1, 1)
P = (⇢, ✓) = (�p2, 5⇡/4)
P = (�p2, 5⇡/4) = (p2, 5⇡/4 + ⇡) =
(
p
2, 9⇡/4) = (
p
2, ⇡/4)
x = �p2 cos 5⇡/4 = p2 cos⇡/4 = 1
y = �p2 sen 5⇡/4 = p2 sen⇡/4 = 1
P
⇤
P = (4, 5)
P = (
p
41, ✓0)
P = (x, y) = (4, 5)
x = 4 y = 5
⇢ =
p
42 + 52 =
p
16 + 25 =
p
41
cos ✓0 =
4p
41
sen ✓0 =
5p
41
(⇢, ✓) = (
p
41, ✓0) = (�
p
41, ✓0 + ⇡)
P
⇤
P = (0,�4)
P = (�4,⇡/2)
P = (x, y) = (0,�4)
x = 0 y = �4
⇢ =
p
02 + (�4)2 = p16 = 4
cos ✓ =
0
4
= 0 sen ✓ =
�4
4
= �1
(⇢, ✓) = (4, 3⇡/2) = (�4, 3⇡/2 + ⇡) =
(�4, 5⇡/2) = (�4, ⇡/2) P
⇤
(x, y)
(⇢, ✓)
P Q
r
��!
PQ ? r d(P, r) = d(Q, r)
C
• OX8<:(x, y) 2 C () (x,�y) 2 C(⇢, ✓) 2 C () (⇢,�✓) 2 C (�⇢, ⇡ � ✓) 2 C;
�OX
• OY8<:(x, y) 2 C () (�x, y) 2 C(⇢, ✓) 2 C () (⇢, ⇡ � ✓) 2 C (�⇢,�✓) 2 C;
�OY
• y = x8<:(x, y) 2 C () (y, x) 2 C(⇢, ✓) 2 C () (⇢, ⇡2 � ✓) 2 C (�⇢, 3⇡2 � ✓) 2 C;
y = x
• y = �x8<:(x, y) 2 C () (�y,�x) 2 C(⇢, ✓) 2 C () (⇢, 3⇡2 � ✓) 2 C (�⇢, ⇡2 � ✓) 2 C.
y = �x
⇢ = 2
C : ⇢ = 2
⇢ =
p
x2 + y2
⇢ = 2 () px2 + y2 = 2
() x2 + y2 = 4 .
⇢ = 2
2 ⇤
C : ✓ = 3⇡
4
y
x
= tg ✓
✓ =
3⇡
4
() y
x
= tg
3⇡
4
=
sen((3⇡)/4)
cos((3⇡)/4)
=
p
2/2
�p2/2 = �1 .
y
x
= �1 y = �x
✓ = 3⇡4
⇤
C : ⇢ cos(✓ � ⇡/3) = 2 .
cos(a� b) = cos a cos b+ sen a sen b
⇢ cos
⇣
✓ � ⇡
3
⌘
= 2() ⇢ cos ✓ cos
⇣
⇡
3
⌘
+ ⇢ sen ✓ sen
⇣
⇡
3
⌘
= 2 .
x = ⇢ cos ✓ , y = ⇢ sen ✓ , cos
⇣
⇡
3
⌘
=
1
2
sen
⇣
⇡
3
⌘
=
p
3
2
C : x
⇣
1
2
⌘
+ y
✓p
3
2
◆
= 2
C : x+ yp3� 4 = 0 ,
�!v = (1,p3) P = (4, 0)
r : ⇢ cos(✓ � ⇡/3) = 2 r : x+ yp3� 4 = 0
⇤
C : ⇢ cos ✓ = 3
C : ⇢ cos ✓ = 3
x = ⇢ cos ✓ C : x = 3
�OX
(3, 0)
⇤
C : ⇢ = 2b sen ✓ b > 0
C : ⇢ = 2b sen ✓ b > 0
⇢ = ±px2 + y2 sen ✓ = ± yp
x2 + y2
±px2 + y2 = ± 2byp
x2 + y2
() x2 + y2 = 2by
() x2 + y2 � 2by = 0
() x2 + (y � b)2 = b2
C
b (0, b)
⇤
C : ⇢2 � 4⇢ cos ✓ + 2 = 0
⇢2 = x2 + y2 x = ⇢ cos ✓
x2 + y2 � 4x+ 2 = 0() (x� 2)2 + y2 = 2 ,
(2, 0)
p
2
C C1 C2
⇤
C : ⇢ = 2
3� cos ✓
⇢ > 0 ✓ 2 [0, 2⇡] ⇢ = px2 + y2
cos ✓ =
xp
x2 + y2
C
p
x2 + y2 =
2
3� xp
x2 + y2
() 3px2 + y2 � x = 2
() 3px2 + y2 = x+ 2
() 9(x2 + y2) = x2 + 4x+ 4
() 8x2 � 4x+ 9y2 = 4
() 8
⇣
x2 � x
2
⌘
+ 9y2 = 4
() 8
⇣
x� 1
4
⌘2
+ 9y2 = 4 + 8⇥ 1
16
=
9
2
()
✓
x� 1
4
◆2
9
16
+
y2
1
2
= 1
C C C =
⇣
1
4
, 0
⌘
a =
3
4
b =
1p
2
` : y = 0 `0 : x =
1
4
A1 =
⇣
�1
2
, 0
⌘
A2 = (1, 0)
B1 =
✓
1
4
,� 1p
2
◆
B2 =
✓
1
4
,
1p
2
◆
C : ⇢ =
2
3� cos ✓
⇤
C : ⇢ = 1 + sen 2✓
sen 2✓ = 2 sen ✓ cos ✓ ,
⇢ = 1 + 2 sen ✓ cos ✓
⇢ � 0 ✓ 2 R
p
x2 + y2 = 1 +
2xy
x2 + y2
() (x2 + y2)3/2 = x2 + y2 + 2xy = (x+ y)2
C ⇥�⇡4 , ⇡4 ⇤
C
y = x
(x, y) 2 C () (y, x) 2 C
y = �x (x, y) 2 C () (�y,�x) 2 C
⇢ = 1+sen 2✓
✓
h
�⇡
4
,
⇡
4
i
⇢ = 0 ✓ = �⇡
4
⇢ = 1
✓ = 0 ⇢ = 2 ✓ =
⇡
4
⇢ > 0
✓ 2
⇣
�⇡
4
,
⇡
4
i
h
�⇡
4
,
⇡
4
i
C
C : ⇢ = 1 + sen 2✓
⇤
C : ⇢ = 1 + 2 cos ✓
⇢
⇢ = ±px2 + y2 cos ✓ = ±xp
x2 + y2
⇢ ✓
±px2 + y2 = 1± 2xp
x2 + y2
() x2 + y2 = ±px2 + y2 + 2x
() (x2 + y2 � 2x)2 = x2 + y2
�OX �OY
✓ [0, ⇡]
✓ 2 [0, ⇡]
• ⇢ = 1 + 2 cos ✓ = 0 cos ✓ = �1
2
⇢ = 0
✓0 = ⇡ � ⇡3 =
2⇡
3
• ⇢ > 0 �1
2
< cos ✓  1 0  ✓ < 2⇡
3
• ⇢ < 0 �1  cos ✓ < �1
2
2⇡
3
< ✓  ⇡
P1 = (3, 0) P2 = (2, ⇡/3) P3 = (1, ⇡/2) P4 = (0, 2⇡/3)
P5 = (�1, ⇡)
[0, ⇡]
C ✓ [0,⇡]
C
�OX
C ⇤
C : ⇢2 = cos ✓
⇢ = ±px2 + y2 cos ✓ = ±xp
x2 + y2
x2 + y2 =
±xp
x2 + y2
() (x2 + y2)3/2 = ±x() (x2 + y2)3 = x2 .
OX OY
[0, ⇡/2]
⇢ = 0 cos ✓ = 0 ⇢ = 0 ✓ = ⇡/2
✓ 2 [0, ⇡/2]
P1 = (1, 0) P2 =
�
1/21/4 , ⇡/4
�
P3 = (0 , ⇡/2)
C C
OX OY
C ⇤
C : ⇢ = 2 sen2 ✓
2
2 sen2
✓
2
= 1� cos ✓
C : ⇢ = 1� cos ✓ .
⇢ � 0 ⇢ =px2 + y2 cos ✓ = xp
x2 + y2p
x2 + y2 = 1� xp
x2 + y2
() x2 + y2 =px2 + y2 � x
() x2 + y2 + x =px2 + y2
C
C �OX
�OY [0, ⇡]
P1 = (0, 0) P2 = (1, ⇡/2) P3 = (2, ⇡) C
[0, ⇡] C
C
C �OX
C
⇤
C : ⇢ = cos 2✓
⇢ = ±px2 + y2 cos 2✓ = cos2 ✓ � sen2 ✓ = x2 � y2
x2 + y2
±px2 + y2 = x2 � y2
x2 + y2
() ±(x2 + y2)3/2 = x2 � y2
() (x2 + y2)3 = (x2 � y2)2
OX
OY y = x y = �x h
0,
⇡
4
i
• ⇢ > 0 ✓ 2
h
0,
⇡
4
⌘
C ✓⇥
0, ⇡4
⇤
• ⇢ = cos 2✓ = cos ⇡
2
= 0 ✓ =
⇡
4
• ⇢ = cos 2✓ = cos 0 = 1 ✓ = 0
✓ [0, ⇡/4]
OX OY y = x
C
⇤
C : ⇢ = sen 3✓
sen 3✓ = sen(✓ + 2✓) = sen ✓ cos 2✓ + cos ✓ sen 2✓
= sen ✓(cos2 ✓ � sen2 ✓) + 2 sen ✓ cos2 ✓ = 3 sen ✓ cos2 ✓ � sen3 ✓
= sen ✓(3 cos2 ✓ � sen2 ✓),
±
p
x2 + y2 =
±yp
x2 + y2
✓
3x2 � y2
x2 + y2
◆
() (x2 + y2)2 = y(3x2 � y2)
�OY
�OX
✓ [0, 2⇡]
• ⇢ = 0() sen 3✓ = 0() 3✓ = 0, ⇡, 2⇡, 3⇡, 4⇡, 5⇡, 6⇡ () ✓ = 0, ⇡
3
,
2⇡
3
, ⇡,
4⇡
3
,
5⇡
3
, 2⇡ ;
• ⇢ = 1() sen 3✓ = 1() 3✓ = ⇡
2
, 2⇡ +
⇡
2
, 4⇡ +
⇡
2
() ✓ = ⇡
6
,
5⇡
6
,
9⇡
6
;
• ⇢ = �1() sen 3✓ = �1() 3✓ = 3⇡
2
, 2⇡+
3⇡
2
, 4⇡+
3⇡
2
() ✓ = ⇡
2
,
7⇡
6
,
11⇡
6
• ⇢ > 0
⇣
0,
⇡
3
⌘
[
⇣
2⇡
3
, ⇡
⌘
[
⇣
4⇡
3
,
5⇡
3
⌘
• ⇢ < 0
⇣
⇡
3
,
2⇡
3
⌘
[
⇣
⇡,
4⇡
3
⌘
[
⇣
5⇡
3
, 2⇡
⌘
C
⇤
⇢ � 0
R = R1[R2
R1 :
8<:0  ⇢ 
2
cos ✓
�⇡
4
 ✓  0
R2 :
8<:2 sen ✓  ⇢ 
2
cos ✓
0  ✓  ⇡
4
(⇢, ✓) R
⇢ =
2
cos ✓
() ⇢ cos ✓= 2() x = 2
⇢ = 2 sen ✓ () ±
p
x2 + y2 =
±2yp
x2 + y2
() x2 + y2 = 2y ()
x2 + (y � 1)2 = 1 (0, 1) 1
✓ =
⇡
4
() y
x
= tg ✓ = 1 () y = x
✓ = �⇡
4
() y
x
= tg ✓ = �1() y = �x
R
OXY
R :
8>><>>:
x2 + y2 � 2y � 0
x  2
x� y � 0
x+ y � 0
x2 + y2 = 2y y = x
(0, 0) (1, 1) x2+y2 = 2y y = 1�p1� x2 y 2 [0, 1]
x 2 [0, 1] R S1 [ S2
S1 :
8<:�x  y  1�
p
1� x2
0  x  1
S2 :
8<:�x  y  x1  x  2 .
⇤
(
⇢1(✓)  ⇢  ⇢2(✓)
✓1  ✓  ✓2
.
R
C1 : (x�2)2+y2 = 4
C2 : y = 1 C3 : x� y = 0 C4 : y = 0 R
(x�2)2+y2 = 4() x2�4x+4+y2 = 4() x2+y2 = 4x() ⇢2 = 4⇢ cos ✓
() ⇢ = 4 cos ✓
y = 1() ⇢ sen ✓ = 1() ⇢ = 1
sen ✓
x� y = 0() x = y () tg ✓ = 1() ✓ = ⇡
4
y = 0() ⇢ sen ✓ = 0() sen ✓ = 0() ✓ = 0
C2 \ C3 = {(1, 1)} C1 \ C2 = {(2 �
p
3, 1), (2 +
p
3, 1)} y = ±p4x� x2
x = 2±
p
4� y2 (x, y) 2 C1
R :
8<:0  ⇢  4 cos ✓0  ✓  ✓0 S
8<:0  ⇢  1/sen ✓✓0  ✓  ⇡/4
tg ✓0 =
1
2 +
p
3
= 2 � p3
✓0 2
⇣
0,
⇡
2
⌘
R :
8<:0  y  x0  x  1 S
8<:0  y  11  x  2 +p3 S
8<:0  y 
p
4x� x2
2 +
p
3  x  4
R :
8<:y  x  2 +
p
4� y2
0  y  1
⇤
R
C1 : x2 + y2 = 2 C2 : y = x2
C1 : ⇢ =
p
2 C2 : ⇢ sen ✓ = ⇢2 cos2 ✓
C2 : ⇢ = tg ✓ sec ✓
C1\C2 = {(1, 1), (�1, 1)} ✓h
�⇡ � ⇡
4
,
⇡
4
i
=
h
�5⇡
4
,
⇡
4
i
R :
8<:tg ✓ sec ✓  ⇢ 
p
2
�5⇡
4
 ✓  �⇡
S 8<:0  ⇢ 
p
2
�⇡  ✓  0
S 8<:tg ✓ sec ✓  ⇢ 
p
2
0  ✓  ⇡
4
R :
8<:�
p
2� x2  y  x2
�1  x  1S8<:�
p
2� x2  y  p2� x2
�p2  x  �1S8<:�
p
2� x2  y  p2� x2
1  x  p2
⇤
R C1 : ⇢ = 4
p
3 cos ✓
C2 : ⇢ = 4 sen ✓
• C1 : ⇢ = 4
p
3 cos ✓ () ±
p
x2 + y2 = 4
p
3
✓ ±xp
x2 + y2
◆
() x2+y2 = 4p3 x
() (x� 2p3)2 + y2 = 12 (2p3, 0) 2p3
• C2 : ⇢ = 4 sen ✓ () ±
p
x2 + y2 = 4
✓ ±yp
x2 + y2
◆
() x2 + y2 = 4y
() x2 + (y � 2)2 = 4 (0, 2) 2
R
OXY
(x, y) 2 C1 \ C2 () x2 + y2 = 4
p
3 x x2 + y2 = 4y
() y = p3x x2 + y2 = 4y
() y = p3x x2 + 3x2 = 4p3x
() y = p3x 4x2 = 4p3 x
() x = 0 y = 0 x = p3 y = 3 .
C1 \ C2 =
n
(0, 0),
⇣p
3, 3
⌘o
✓0 OP0 P0 =
⇣p
3, 3
⌘
�OX
⇡
3
tg ✓0 =
y
x
=
p
3
R = R1 [R2
R1 :
8<:0  ⇢  4 sen ✓0  ✓  ⇡
3
R2 :
8<:0  ⇢  4
p
3 cos ✓
⇡
3
 ✓  ⇡
2
,
R :
(
2
p
3�
p
12� y2  x 
p
4� (y � 2)2
0  y  3 .
⇤
R
R :
8<:
x2
12
 y  1
2
p
16� x2
0  x  2p3 .
R
R :
(
⇢1(✓)  ⇢  ⇢2(✓)
✓1  ✓  ✓2 ,
(⇢, ✓)
R
• x = 0 x = 2p3
• C1 : x2 = 12y �OY
• C2 y � 0
C2 : 2y =
p
16� x2 =) 4y2 = 16� x2 =) x2 + 4y2 = 16 =) x
2
16
+
y2
4
= 1 ,
C = (0, 0) (4, 0) (�4, 0) (0, 2) (0,�2)
�OX ⇣
2
p
3, 1
⌘
2 C1 \ C2 R
R
C1 C2
• 12y = x2 () 12⇢ sen ✓ = ⇢2 cos2 ✓ () ⇢ = 12 sen ✓
cos2 ✓
= 12 tg ✓ sec ✓
• x2+4y2 = 16 () ⇢2(cos2 ✓+4 sen2 ✓) = 16 () ⇢2(1�sen2 ✓+4 sen2 ✓) = 16
() ⇢ = 4p
1 + 3 sen2 ✓
✓0 2
⇣
0,
⇡
2
⌘
tg ✓0 =
1
2
p
3
=
p
3
6
R = R1 [R2
R1 :
8><>: 0  ⇢  12 tg ✓ sec ✓0  ✓  ✓0 R2 :
8>><>>:
0  ⇢  4p
1 + 3 sen2 ✓
✓0  ✓  ⇡2 .
⇤