Question

What is the distance between the following polar coordinates?: `(2,(5pi)/12), (1,(3pi)/12)`

Answer 1

`D=sqrt(5-2sqrt3)~~1.2393`

Explanation:

We know that ,

`"Distance between Polar Co-ordinates:"A(r_1,theta_1)and B(r_2,theta_2)` is

`color(red)(D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))...to(I)`

We have , `P_1(2,(5pi)/12) and P_2(1,(3pi)/12)`.

So , `r_1=2 , r_2=1 , theta_1=(5pi)/12 and theta_2=(3pi)/12`

`=>theta_1-theta_2=(5pi)/12-(3pi)/12=(2pi)/12=(pi)/6=30^circ`

`=>cos(theta_1-theta_2)=cos(30^circ)`

`"Using : " color(red)((I)` we get

`D=sqrt(2^2+1^2-2(2)(1)cos30^circ)`

`=>D=sqrt(4+1-4*sqrt3/2)`

`=>D=sqrt(5-2sqrt3)`

`=>D~~1.2393`