# How do you convert # (r-1)^2= - sin theta costheta +cos^2theta# to Cartesian form?

We know the relations

Given equation

This equation represents the cartesian form of the given polar equation.

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graph{(x^2+y^2)(x^2+y^2-2sqrt(x^2+y^2))+y(x+y)=0 [-5, 5, -2.5, 2.5]}

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To convert the equation ( (r - 1)^2 = -\sin(\theta)\cos(\theta) + \cos^2(\theta) ) from polar form to Cartesian form, we'll use the identities ( r^2 = x^2 + y^2 ), ( \sin(\theta) = \frac{y}{r} ), and ( \cos(\theta) = \frac{x}{r} ).

First, we'll square both sides of ( r - 1 ):

[ (r - 1)^2 = r^2 - 2r + 1 ]

Then, we'll substitute ( -\sin(\theta)\cos(\theta) + \cos^2(\theta) ) with their corresponding Cartesian form equivalents:

[ -\sin(\theta)\cos(\theta) + \cos^2(\theta) = -\frac{y}{r}\cdot\frac{x}{r} + \left(\frac{x}{r}\right)^2 ]

[ = -\frac{xy}{r^2} + \frac{x^2}{r^2} = \frac{x^2 - xy}{r^2} ]

Substituting these expressions back into the original equation, we get:

[ r^2 - 2r + 1 = \frac{x^2 - xy}{r^2} ]

Now, we'll replace ( r^2 ) with ( x^2 + y^2 ):

[ x^2 + y^2 - 2\sqrt{x^2 + y^2} + 1 = \frac{x^2 - xy}{x^2 + y^2} ]

[ x^2 + y^2 - 2\sqrt{x^2 + y^2} + 1 = \frac{x^2 - xy}{x^2 + y^2} ]

[ (x^2 + y^2 - 1) = 2\sqrt{x^2 + y^2} - \frac{xy}{x^2 + y^2} ]

This is the equation in Cartesian form.

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To convert the equation ( (r-1)^2 = -\sin(\theta)\cos(\theta) + \cos^2(\theta) ) from polar form to Cartesian form, you can use the trigonometric identities ( r = \sqrt{x^2 + y^2} ), ( \sin(\theta) = \frac{y}{r} ), and ( \cos(\theta) = \frac{x}{r} ).

Substituting these identities into the given equation and simplifying yields:

[ (x^2 + y^2 - 2x + 1) = -\frac{y}{r}\frac{x}{r} + \frac{x^2}{r^2} ]

After substituting ( r^2 = x^2 + y^2 ), we simplify further:

[ x^2 + y^2 - 2x + 1 = -\frac{xy}{x^2 + y^2} + \frac{x^2}{x^2 + y^2} ]

Then, multiplying both sides by ( x^2 + y^2 ) gives:

[ (x^2 + y^2)^2 - 2x(x^2 + y^2) + (x^2 + y^2) = -xy + x^2 ]

Expanding and rearranging terms gives the final Cartesian form:

[ x^4 - 2x^3 + x^2 + y^2(x^2 - 2x + 1) = -xy ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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