1. convert polar equation to rectangular coordinates:

r^2=4sin2theta

2. find the coefficient of x^5 in the expansion (x+2)(x+1)^8

To convert a polar equation to rectangular coordinates, you can use the following conversion formulas:

x = r * cos(theta)
y = r * sin(theta)

Let's use these formulas to convert the provided polar equation to rectangular coordinates:

1. Convert polar equation to rectangular coordinates:
r^2 = 4sin(2theta)

We can start by rewriting the equation using the double-angle trigonometric identity:
r^2 = 4sin^2(theta) * 2cos(theta)

Now, replace sin^2(theta) with (1 - cos^2(theta)):
r^2 = 4(1 - cos^2(theta)) * 2cos(theta)

Simplifying the equation further:
r^2 = 8cos(theta) - 8cos^3(theta)

Now, use the conversion formulas to express r in terms of x and y:
r^2 = x^2 + y^2
cos(theta) = x / r

Substituting these expressions into the equation:
x^2 + y^2 = 8(x / sqrt(x^2 + y^2)) - 8(x / sqrt(x^2 + y^2))^3

Now, we can simplify this equation further to get the rectangular form of the equation.