Question

What is the distance between `(1 ,( pi)/4 )` and `(4 , ( 3 pi )/2 )`?

Answer 1

The first coordinate is from the unit circle (`pi/4=45^o`) and the second coordinate lies directly on the y-axis (`(3pi)/2=270^o`)

Explanation:

`(1,pi/4)` from the unit circle is `(sqrt2/2, sqrt2/2)`

`(4,(3pi)/2)` on the y-axis `(-4,0)`

Using the distance formula ...

distance `=sqrt[(-4-sqrt2/2)^2+((0-sqrt2/2)^2]`

`~~4.7599`

hope that helped

Answer 2

`4.761` units

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 `(1,(pi)/4)` and `(r_2,theta_2)` represent `(4,(3pi)/2)`.
`implies d=sqrt(1^2+4^2-2*1*4Cos((pi)/4-(3pi)/2)`
`implies d=sqrt(1+16-8Cos((-5pi)/4)`
`implies d=sqrt(17-8*(-0.7085))=sqrt(17+5.668)=sqrt(22.668)=4.761` units
`implies d=4.761` units (approx)
Hence the distance between the given points is `4.761` units.