POLAR COORDINATES

POLAR COORDINATES

Polar points are plotted using the polar coordinate plane with the line OX as initial line or polar axis, point 0 as the pole or origin and the distance of the point from O as the radius vector.

The position of any point P in the plane is determined if the length of the line OP together with the angle that this line makes with OX are known, both the length and the angle being measured in a definite sense.

From the figure :

r = radius vector,

+ if measured along the terminal side of Ө,

– if measured in the opposite direction along the terminal side of Ө

Ө = polar angle

OX = initial line or polar axis

O = pole or origin

Polar Coordinate paper – it is where the polar points are plotted.

( to be illustrated on the board during lecture )

Plot the following points :

1. ( 2, 30º )                 6. ( 3, 60º )

2. ( 4, 225º )                7. ( 4, – 315º )

3. ( – 6, –120º )             8. ( – 3, –240º )

4. ( – 4,  330º )             9. ( 4, – 330º )

5. ( 6, – 150º )             10. (– 6, 150º )

Distance between two points in Polar Coordinates : by Cosine law

Conversion

A. Polar to rectangular coordinates

1) x = r cos Ө

2) y = r sine Ө

B. Rectangular to Polar coordinates

   1)  r = sqrt( x² + y² )

   2)   Ө = Arctan ( y/x )

Exercises :

1. The point (r, Ө) is equidistant from (2, 90º ) and (– 2, 150º). Express the statement into an algebraic expression.

2. Show that the given points ( 2 , 45º ), ( sqrt( 2 ), 90º ) and ( – 2 , 135º ) are vertices of a right triangle and find its area.

3. Convert to rectangular coordinate.

a) ( 3 , 240º )

b) ( 4 , 150º )

c) (– 5 , 150º )

d) ( – 3 ,  – 150º )
e)  (  5 ,  – 150º )

4. Convert to polar coordinate

a) (2, –2 )

b) (– 1, –sqrt(3))

c) (– 3, 3)

5. Find the distance between the following points.

a) ( 3 , 240º ) and (– 5 , 150º )

b) ( 4 , 150º ) and ( 2 , 45º )

c) ( 2 , 90º ) and ( – 2 , 135º )

d)  ( – 3 , 240º ) and ( 5 ,  – 150º )
e)  ( – 2 , 90º ) and (  2 ,  – 135º )