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.