How do you convert #r^2 = sin 2(theta)# into cartesian form?
The equation is
The equation is
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To convert (r^2 = \sin(2\theta)) into Cartesian form:

Recall the relationship between polar coordinates ((r, \theta)) and Cartesian coordinates ((x, y)): (x = r \cos(\theta)) (y = r \sin(\theta))

Square both sides of the equation (r^2 = \sin(2\theta)) to get rid of the square root: ((r^2)^2 = (\sin(2\theta))^2)

Substitute (r^2 = x^2 + y^2) and (\sin(2\theta) = 2 \sin(\theta) \cos(\theta)) into the equation: ((x^2 + y^2)^2 = (2 \sin(\theta) \cos(\theta))^2)

Expand both sides: ((x^2 + y^2)^2 = 4 \sin^2(\theta) \cos^2(\theta))

Use trigonometric identities: (\sin^2(\theta) = 1  \cos^2(\theta)) and (\cos^2(\theta) = 1  \sin^2(\theta))

Substitute the identities into the equation: ((x^2 + y^2)^2 = 4(1  \cos^2(\theta))(1  \sin^2(\theta)))

Expand and simplify: ((x^2 + y^2)^2 = 4(1  \cos^2(\theta)  \sin^2(\theta) + \sin^2(\theta)\cos^2(\theta)))

Replace (\cos^2(\theta) + \sin^2(\theta)) with 1: ((x^2 + y^2)^2 = 4(1  1 + \sin^2(\theta)\cos^2(\theta)))

Simplify further: ((x^2 + y^2)^2 = 4(\sin^2(\theta)\cos^2(\theta)))

Take the square root of both sides: (x^2 + y^2 = 2\sin(\theta)\cos(\theta))

Substitute (x = r \cos(\theta)) and (y = r \sin(\theta)) back into the equation: ((r \cos(\theta))^2 + (r \sin(\theta))^2 = 2\sin(\theta)\cos(\theta))

Simplify: (r^2(\cos^2(\theta) + \sin^2(\theta)) = 2\sin(\theta)\cos(\theta))

Since (\cos^2(\theta) + \sin^2(\theta) = 1): (r^2 = 2\sin(\theta)\cos(\theta))
So, the Cartesian form of (r^2 = \sin(2\theta)) is (r^2 = 2\sin(\theta)\cos(\theta)).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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