Question 6.3.3: Polar-to-Rectangular Point Conversion Find the rectangular c...
Polar-to-Rectangular Point Conversion
Find the rectangular coordinates of the points with the following polar coordinates:
a. (2, \frac{3π}{2}) b. (-8, \frac{π}{3}).
We find (x, y) by substituting the given values for r and θ into x = r cos θ and y = r sin θ.
a. We begin with the rectangular coordinates of the point (r, θ) = (2, \frac{3π}{2}).
x = r \cos θ = 2 \cos \frac{3π}{2} = 2 · 0 = 0
y = r \sin θ = 2 \sin \frac{3π}{2} = 2(-1) = -2
The rectangular coordinates of (2, \frac{3π}{2}) are (0, -2). See Figure 6.25.
b. We now find the rectangular coordinates of the point (r, θ) = (-8, \frac{π}{3}).
x = r \cos θ = -8 \cos \frac{π}{3} = -8 (\frac{1}{2}) = -4
y = r \sin θ = -8 \sin \frac{π}{3} = -8 (\frac{\sqrt{3}}{2}) = -4\sqrt{3}
The rectangular coordinates of (-8, \frac{π}{3}) are (-4, -4\sqrt{3}).
