Question

What is the polar form of `( 13,-4 )`?

Answer 1

Polar form is `(13.6 , 5.985) or (13.6 , -0.298)`

Explanation:

`(13 , -4)` lies on `4` th quadrant.

`r =sqrt(13^2 + (-4)^2)= sqrt 185=13.6`

`tan alpha = 4/13 :. alpha = tan^-1(4/13)=0.298498`

Since `theta` lies `4` th quadrant.

`theta= 2pi- alpha = 2pi- 0.298498 ~~5.984686` or

`theta= (- alpha)= -0.298498`

Polar form is `(r, theta) :. (13.6 , 5.985) or (13.6 , -0.298)` [Ans]

Answer 2

`(sqrt(185) , -(19pi)/200 )`

Explanation:

Polar coordinate form is : `( r , theta )`

Cartesian coordinate form `( x , y )`

`x = rcostheta`

`y =rsintheta`

`theta= arctan(y/x)`

`:.`

`13=rcosthetacolor(white)(88)[1]`

`-4=rsinthetacolor(white)(88)[2]` Squaring:

`169=r^2cos^2theta`

`16=r^2sin^2thetacolor(white)(88)` adding [1] and [2]

`185=r^2(sin^2theta+cos^2theta)` `color(white)(88888)sin^2theta+cos^2theta=1`

`:.`

`185=r^2=>r=sqrt(185)`

`tantheta=-4/13=>theta=tan^-1(-4/13)~~-0.2985~~-(19pi)/200`

`:.`

`(sqrt(185) , -(19pi)/200 )`