determine the rectangular coordinates for each of the giving polar cordinates.

Question

determine the rectangular coordinates for each of the giving polar coordinate.

 

1.(1,pi/2)

 

2.(2,5pi/4)

 

And rectangular to polar

 

1.(-5,-5√3)

 

2.(-6,0)

Answer 1

For converting from Polar coordinates (r,θ) to Rectangular coordinates (X,Y) use the following relations:

 

X  = r * Cos(θ)

 

Y  = r * Sin(θ)

                    

Then the following solutions are:

 

1.  r =1   θ = pi/2     X = (1)*Cos(pi/2) = 0     Y = (1)*Sin(pi/2) = 1

 

2.  r = 2.5  θ = pi/4     X = (2.5)*Cos(pi/4)= 1.76    Y = (2.5)*Sin(pi/4) = 1.76

 

 

For converting from Rectangular coordinates (X, Y) to Polar coordinates (r,θ), use the following relations:

 

Tan (θ) = Y/X     ⇒  θ = atan (Y/X)

 

r  = sqrt( X^2 + Y^2)

 

Solutions:

 

3. X = -5   Y = -5√3         Tan(θ) = -5√3/(-5) = 1.732    ⇒  θ = pi/3    r = sqrt( 25 + 75)  = 10   Polar: (10, pi/3)

 

4. X = -6 Y = 0                Tan(θ) = 0/(-6) = 0   ⇒ θ = 0       r = sqrt( 36 + 0) = 6   Polar: (6,0)