# How do you convert #r = 1-2 cosθ# into rectangular forms?

a = 1; b = -2

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To convert the polar equation ( r = 1 - 2\cos(\theta) ) into rectangular form, we can use the relationships between polar and rectangular coordinates:

( x = r \cos(\theta) ) ( y = r \sin(\theta) )

Substituting ( r = 1 - 2\cos(\theta) ) into these equations:

( x = (1 - 2\cos(\theta)) \cos(\theta) ) ( y = (1 - 2\cos(\theta)) \sin(\theta) )

We then simplify these expressions:

( x = \cos(\theta) - 2\cos^2(\theta) ) ( y = \sin(\theta) - 2\sin(\theta)\cos(\theta) )

Using the trigonometric identity ( \cos^2(\theta) = 1 - \sin^2(\theta) ), we can rewrite ( \cos^2(\theta) ) as ( 1 - \sin^2(\theta) ):

( x = \cos(\theta) - 2(1 - \sin^2(\theta)) ) ( y = \sin(\theta) - 2\sin(\theta)\cos(\theta) )

Further simplifying:

( x = \cos(\theta) - 2 + 2\sin^2(\theta) ) ( y = \sin(\theta) - 2\sin(\theta)\cos(\theta) )

Using the trigonometric identity ( 2\sin^2(\theta) = 1 - \cos(2\theta) ):

( x = \cos(\theta) - 2 + 1 - \cos(2\theta) ) ( y = \sin(\theta) - 2\sin(\theta)\cos(\theta) )

Simplifying further:

( x = -1 + \cos(\theta) - \cos(2\theta) ) ( y = \sin(\theta) - 2\sin(\theta)\cos(\theta) )

These are the rectangular forms of the given polar equation.

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