Q 1 . Change the following polar equations into Cartesian equations:
(a) r = a sin 2 A
(b) A=π/4
soliction
(a) r = a sin 2 A
(b) A=π/4soliction
To change a polar equation to a Cartesian equation, we can use the relationship between the polar coordinates (r, θ) and the Cartesian coordinates (x, y), which are given by:
x = r * cos(θ)
y = r * sin(θ)
Let's apply this relationship to the given polar equations:
(a) r = a sin(2θ)
To express this equation in terms of Cartesian coordinates (x, y), we substitute the values of r and θ using the equations mentioned above:
x = (a sin(2θ)) * cos(θ)
y = (a sin(2θ)) * sin(θ)
Simplifying these equations, we get:
x = a sin(2θ) * cos(θ)
y = a sin(2θ) * sin(θ)
(b) θ = π/4
To convert this polar equation to a Cartesian equation, we substitute the value of θ in terms of Cartesian coordinates:
x = r * cos(π/4)
y = r * sin(π/4)
Simplifying further:
x = r * (1/sqrt(2))
y = r * (1/sqrt(2))
Therefore, the Cartesian equations are:
(a) x = a sin(2θ) * cos(θ)
y = a sin(2θ) * sin(θ)
(b) x = r * (1/sqrt(2))
y = r * (1/sqrt(2))
Note: In (b), we still have the variable "r" remaining, so in order to fully convert it into Cartesian form, we would need additional information or context.
If you are using A for θ, then clearly θ=π/4 is a straight line with slope 1, so that is just
y = x
r = a sin2θ
r = 2a sinθ cosθ
r^3 = 2a rsinθ rcosθ
(x^2+y^2)^(3/2) = axy
But that only gives half of the graph, where xy is positive.