Question

What is the distance between `(2 ,(7 pi)/6 )` and `(3 , (- pi )/8 )`?

Answer 1

`1.0149`

Explanation:

The distance formula for polar coordinates is

`d=sqrt(r_1^2+r_2^2-2r_1r_2Cos(theta_1-theta_2)`
Where `d` is the distance between the two points, `r_1`, and `theta_1` are the polar coordinates of one point and `r_2` and `theta_2` are the polar coordinates of another point.
Let `(r_1,theta_1)` represent `(2,(7pi)/6)` and `(r_2,theta_2)` represent `(3,-pi/8)`.
`implies d=sqrt(2^2+3^2-2*2*3Cos((7pi)/6-(-pi/8))`
`implies d=sqrt(4+9-12Cos((7pi)/6+pi/8)`
`implies d=sqrt(13-12cos((28pi+3pi)/24))=sqrt(13-12cos((31pi)/24))=sqrt(13-12cos(4.0558))=sqrt(13-12*0.9975)=sqrt(13-12*0.9975)=sqrt(13-11.97)=sqrt(1.03)=1.0149` units
`implies d=1.0149` units (approx)
Hence the distance between the given points is `1.0149`.