Change the following polar equations into Cartesian equations:
(a) r = a sin 2 A
(b) A=π/4
To change a polar equation into a Cartesian equation, we need to use the relations between Cartesian (x, y) and polar (r, θ) coordinates.
First, let's address equation (a):
(a) r = a sin 2A
To convert this equation to Cartesian coordinates, we can use the following relations:
x = r * cos(θ)
y = r * sin(θ)
Here, θ is the polar angle.
In the given equation, we have r = a sin(2A). We need to find x and y in terms of a and A.
x = r * cos(θ) = (a sin(2A)) * cos(A)
y = r * sin(θ) = (a sin(2A)) * sin(A)
Thus, the Cartesian equation is:
x = a sin(2A) * cos(A)
y = a sin(2A) * sin(A)
Moving on to equation (b):
(b) A = π/4
This equation gives us the value of the polar angle θ. To convert it to Cartesian coordinates, we can use the relations mentioned above:
x = r * cos(θ)
y = r * sin(θ)
Substituting the given value, θ = π/4, into the Cartesian equations, we get:
x = r * cos(π/4)
y = r * sin(π/4)
Simplifying further, we know that cos(π/4) = sin(π/4) = √2/2:
x = r * (√2/2)
y = r * (√2/2)
So, the Cartesian equation is:
x = (√2/2) * r
y = (√2/2) * r