How would I rewrite this in rectangular form? //
r = 8 sin theta - 2 cos theta?
I apologize, there was an error before.
so, you couldn't fix it and finish it off? The method was right in front of you ...
r = 8sinθ - 2cosθ
r^2 = 8r sinθ - 2r cosθ
x^2+y^2 = 8y-2x
That's ok as an equation, but I suspect you want a more standard form, so keep working it, completing the squares:
x^2+2x + y^2-8y = 0
x^2+2x+1 + y^2-8y+16 = 1+16
(x+1)^2 + (y-4)^2 = 17
To rewrite the equation in rectangular form, we can use the trigonometric identity:
r = √(x^2 + y^2)
x = r cos(theta)
y = r sin(theta)
Let's substitute these equations into the given equation:
r = 8 sin(theta) - 2 cos(theta)
√(x^2 + y^2) = 8 sin(theta) - 2 cos(theta)
x = r cos(theta) = (√(x^2 + y^2)) cos(theta)
y = r sin(theta) = (√(x^2 + y^2)) sin(theta)
Now, let's square both sides of both equations to eliminate the square root:
x^2 + y^2 = (8 sin(theta) - 2 cos(theta))^2
Expanding the equation:
x^2 + y^2 = 64 sin^2(theta) - 32 sin(theta) cos(theta) + 4 cos^2(theta)
Simplifying further:
x^2 + y^2 = 4(16 sin^2(theta) - 8 sin(theta) cos(theta) + cos^2(theta))
Since sin^2(theta) + cos^2(theta) = 1, we can replace it in the equation:
x^2 + y^2 = 4(16 - 8 sin(theta) cos(theta))
Finally, simplifying:
x^2 + y^2 = 64 - 32 sin(theta) cos(theta)
This is the equation in rectangular form.
No problem! To rewrite the equation r = 8 sin(theta) - 2 cos(theta) in rectangular form, we can make use of the following trigonometric identities:
sin(theta) = y/r
cos(theta) = x/r
Given that r represents the radius and theta represents the angle in polar coordinates, we can replace sin(theta) and cos(theta) in the equation with their equivalent expressions in rectangular form:
r = 8 (y/r) - 2 (x/r)
To eliminate the division by r on both sides of the equation, we can multiply through by r:
r^2 = 8y - 2x
Now, we can rearrange the terms to obtain the rectangular form:
r^2 + 2x = 8y
So, the equation in rectangular form is x + (r^2)/2 = 4y.