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
Anyone please help
x^2 + y^2 = r^2
x = rcosθ
y = rsinθ
r^2 = 4 sin2θ
r^2 = 8sinθcosθ
r^2 = 8 * y/r * x/r
r^4 = 8xy
(x^2 + y^2)^2 - 8xy = 0
(x+1)^8 = x^8 + 8x^7 + 28x^6 + 56x^5 + 70x^4 + ...
(x+2)(x+1)^8 = ... + 70x^5 + ... + 112x^5 + ...
= ... + 182x^5 + ...
1. To convert a polar equation to rectangular coordinates, we can use the following relations:
x = r * cos(theta)
y = r * sin(theta)
Given the polar equation r^2 = 4sin(2theta), we can substitute these relations to get:
x^2 + y^2 = 4sin(2theta)
Now we'll convert sin(2theta) to its rectangular form using the double angle formula:
sin(2theta) = 2sin(theta)cos(theta)
Substituting this back into our equation, we get:
x^2 + y^2 = 4(2sin(theta)cos(theta))
Simplifying further, we have:
x^2 + y^2 = 8sin(theta)cos(theta)
2. To find the coefficient of x^5 in the expansion of (x + 2)(x + 1)^8, we can use the binomial theorem. According to the binomial theorem, the coefficient of x^k in the expansion of (a + b)^n is given by the formula:
C(n, k) * a^(n-k) * b^k
Where C(n, k) represents the binomial coefficient, which is given by:
C(n, k) = n! / (k!(n-k)!)
In our case, a = x+2, b = x+1, n = 8, and k = 5. So the coefficient of x^5 would be:
C(8, 5) * (x+2)^(8-5) * (x+1)^5
Now we need to substitute the values and calculate the binomial coefficient:
C(8, 5) = 8! / (5!(8-5)!)
C(8, 5) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Substituting this back into the original equation, we get:
56 * (x+2)^(8-5) * (x+1)^5
So the coefficient of x^5 in the expansion is 56.