Convert the polar equation to rectangular form.
θ = 4π/3
this is just a straight line at an angle of 4π/3, right?
tanθ = √3
y/x = √3
y = √3 x
To convert a polar equation to rectangular form, we can use the following relationships:
x = r * cos(θ)
y = r * sin(θ)
In this case, we have the polar equation θ = 4π/3. To convert it to rectangular form, we need to find the corresponding values for x and y.
First, let's find the value for r. In a polar equation of the form θ = constant, r represents the distance from the origin to the point on the graph. Since θ = 4π/3, we can see that the angle is fixed at 4π/3 radians.
To find r, we can consider that any point on the graph of θ = 4π/3 will have the same angle of 4π/3, but different distances from the origin. Therefore, we can choose any value for r, as long as it is not zero.
Let's choose r = 1 for simplicity. Now, we can substitute r = 1 and θ = 4π/3 into the equations for x and y:
x = r * cos(θ) = 1 * cos(4π/3) = -1/2
y = r * sin(θ) = 1 * sin(4π/3) = -√3/2
So, the rectangular form of the polar equation θ = 4π/3 is x = -1/2 and y = -√3/2.