Write the following numbers in the polar form r e i θ , 0 ≤ θ < 2 π . (a) 4 r = , θ = , (b) 9 i r = , θ = , (c) − 1 + 7 i r = , θ =

Question

Write the following numbers in the polar form , .

(a)    

 r=      ,    

(b)     

    

(c)  7i  

 

Answer 1

( a) 4 lies on the x-axis . The rectangular coordinates are (4,0).

To obtain the polar coordinates , you compute r = √ 4x4 + 0x0 = 4 and theta =0 degrees ( or radians)

(b) 9i lies on the y-axis. The rectangular coordinates are ( 0, 9). To obtain the polar coordinates you compute r = √ 0*0 +9*9 = 9 and theta = tan-1( 9/0) = 90 ( π/2 in polar) (c) is -1 +7i . It has rectangular coordinaetes ( -1, 7) and lies in Quadrant II. To obtain the polar coordinate r, you compute r = √ (-1)*(-1) + (7)*(7) = √ 50. To obtain the angle theta, you compute, tan-1 ( 7 / -1) = tan-1 (-7) = - 81.869898 degrees( using calculator) . However, you must add 180, resulting in 98.130102, to normalize the value to the range 0 to 360. Convert to polar obtains 1.7126 radians.

Answer 2

Kianamaria

(a). This point lies on the x axis so θ=0. r is the distance of the point from the origin, so r=4

(b) This point lies on the y axis, so θ = π/2. Distance of the point from the origin r=9

(c) 1+7i . This will be in the 1st quadrant. x=1, y=7, so the distance of the point from the origin

r = √ (1² + 7²).

As for θ, as the point is in the first quadrant, tan θ = y/x = 7/1 to θ= tan ^-1 (7/1). [ make sure your calculator is working in radians]

Mike