Write the polar equation in rectangular form... r=5sintheta
to convert to rectangular form I need an r and a theta
then
x = r cos theta
and
y = r sin theta
r = 5sinθ
r^2 = 5r sinθ
x^2+y^2 = 5y
x^2 + (y - 5/2)^2 = 25/4
To write the polar equation in rectangular form, we can use the following conversion formulas:
x = r * cos(theta)
y = r * sin(theta)
In this case, we are given the polar equation r = 5sin(theta).
Substituting the value of r in the conversion formulas, we have:
x = (5sin(theta)) * cos(theta)
y = (5sin(theta)) * sin(theta)
Now, we need to simplify these expressions by using trigonometric identities.
Recall the double angle identity for sine: sin(2θ) = 2sin(θ)cos(θ).
Using the double angle identity for sine, we can rewrite the equation as:
x = (5sin(theta)) * cos(theta)
y = (5sin(theta)) * sin(theta)
x = (5sin(theta)) * cos(theta)
= (5 * 2sin(θ)cos(θ)) * cos(theta)
= 10sin(theta)cos^2(theta)
y = (5sin(theta)) * sin(theta)
= (5 * 2sin(θ)cos(θ)) * sin(theta)
= 10sin^2(theta)cos(theta)
Thus, the polar equation r = 5sin(theta) in rectangular form is:
x = 10sin(theta)cos^2(theta)
y = 10sin^2(theta)cos(theta)