Transcript Sullivan Algebra and Trigonometry: Section 10.1

Sullivan Algebra and
Trigonometry: Section 10.1
Objectives of this Section
• Plot Points Using Polar Coordinates
• Convert from Polar Coordinates to Rectangular
Coordinates
• Covert from Rectangular Coordinates to Polar
Coordinates
Polar axis
Origin Pole
x
P  r , 
r

O Pole
Polar axis
 
Plot the point  4,  using polar coordinates.
 6
4
  
P   4, 
 6

6
O Pole
Polar axis
Plotting r , , r  0

r
P  r , , r  0
7 

Plot the point   4,
 using polar coordinates.
6 

7
6
4
7 


 P    4,
6 

6
O
A point with polar coordinates r,  also can be
represented by any of the following:
r,  2k  or  r,    2k 
k any integer
The polar coordinates of the pole are 0, ,
where  can be any angle.
Find other polar coordinates r ,  of the point
2, 3 for which
(a) r  0, 2    4
(b) r  0, 0    2
(c) r  0,  2    0
(a) P  2,  3  2   2, 7 3
( b) P   2,  3      2, 4 3
( c) P  2 ,  3  2   2 ,  5 3
Conversion from Polar Coordinates to
Rectangular Coordinates
If P is a point with polar coordinates r, , the
rectangular coordinates x, y of P are given by
x  r cos
r  x y
2
2
y  r sin 
2
y
tan  
x
Find the rectangular coordinates of the points
with the following polar coordinates:
(a) 5,  3
(b)  4 , 5 4 
x  r cos
y  r sin 

1 5
(a) x  r cos   5 cos  5  
3
2 2
 3 5 3

y  r sin   5 sin  5

3
2
2



The rectangular coordinates of the point
5,  3 are 5 2 , 5 3 2.

5
2
2 2
(b) x  r cos   4 cos
 4 

4
2


 2
5
  2 2
y  r sin   4 sin
 4

4
2


The rectangular coordinates of the point
- 4, 5 4 are 2 2 ,2 2 .
Find polar coordinates of a point whose
rectangular coordinates are (-3, 4).
r  x  y  (3)  4
2
(x, y) = (-3, 4)
r

2
2
2
 9  16  25  5
y

1 4
  tan
 tan


x
3 3
     3  2 3
1
A set of polar coordinates for the point  3,4
is (5,2 3). Others: 5,8 3 and - 5,5 3
Transform the equation r  cos  sin from
polar coordinates to rectangular coordinates.
r  cos  sin 
2
r  r cos  r sin 
2
2
x  y  x y
x x y  y0
2
2
x  x 1/ 4  y  y 1/ 4  1/ 4 1/ 4
2
2
x  1 / 2    y  1 / 2   1 / 2
2
2
Transform the equation x  3 y from
2
rectangular coordinates to polar coordinates.
x  3y
2
r cos   3r sin 
2
r cos   3r sin   0
2
2