How would I rewrite this in rectangular form? // r = 8 sin theta - 2 sin theta?
I suspect a typo, but
r = 8sinθ - 2sinθ
r = 6sinθ
r sinθ = 6sin^2θ
y = 6(x^2+y^2)
To rewrite the equation in rectangular form, we need to eliminate the trigonometric functions and express the equation in terms of x and y.
First, let's rearrange the given equation:
r = 8 sin(theta) - 2 sin(theta)
Now, we can utilize the trigonometric identity for the coordinate conversion from polar to rectangular form:
x = r * cos(theta)
y = r * sin(theta)
Substituting the given equation into the above identity, we get:
x = (8 sin(theta) - 2 sin(theta)) * cos(theta)
y = (8 sin(theta) - 2 sin(theta)) * sin(theta)
Simplifying further, we have:
x = 6 sin(theta) * cos(theta)
y = 6 sin^2(theta)
Now, we can utilize another trigonometric identity to simplify x:
sin(2theta) = 2sin(theta) * cos(theta)
Therefore, x can be rewritten as:
x = 3 * sin(2theta)
Finally, the equation in rectangular form is:
x = 3 * sin(2theta)
y = 6 * sin^2(theta)
These equations represent the rectangular form of the given polar equation.