convert the polar equation to its rectangular equivalent
r = 4sin (theta)
sin (theta) = y/r
so
r = 4 y / r
[ sqrt (x^2+y^2) ]^2 = 4 y
x^2 + y^2 = 4 y
y^2 - 4 y = -x^2
y^2 -4 y + 4 = -x^2 +4
(y-2)^2 =-(x^2-4)
y-2 = sqrt (4-x^2)
y = 2 - sqrt (4-x^2)
(y-2)^2 =-(x^2-4)
y-2 = or -sqrt (4-x^2)
y = 2 - sqrt (4-x^2)
or y = 2 + sqrt (4-x^2)
r = 4sinθ
r^2 = 4r sinθ
x^2+y^2 = 4y
x^2 + y^2 - 4y = 0
x^2 + y^2 - 4y + 4 = 4
x^2 + (y-2)^2 = 4
To convert a polar equation to its rectangular equivalent, we need to express the values in terms of Cartesian coordinates (x, y).
The polar equation r = 4sin(theta) represents a sinusoidal curve where the distance from the origin (r) is determined by the sine of the angle (theta). To convert this equation to rectangular form, we can use the following relationships:
x = r * cos(theta)
y = r * sin(theta)
Substituting the given equation into these expressions, we get:
x = (4sin(theta)) * cos(theta)
y = (4sin(theta)) * sin(theta)
Now, we can simplify these expressions:
x = 4sin(theta) * cos(theta)
x = 2sin(2theta) [using the double angle identity: sin(2theta) = 2sin(theta) * cos(theta)]
y = 4sin^2(theta)
y = 2(1 - cos(2theta)) [using the identity: sin^2(theta) = (1 - cos(2theta))/2]
Therefore, the rectangular equivalent of the polar equation r = 4sin(theta) is:
x = 2sin(2theta)
y = 2(1 - cos(2theta))