How do you convert #r = 6/(3  4cos(theta))# into cartesian form?
The given equation
This is the cartesian form of the given polar equation.
graph{7x^29y^2+48x+36=0}
The source for what follows is A. S. Adikesavan.
For information, the polar equation
represents
( parabola ellipse hyperbola) according as
For d = 2, a = 2, b = 2, c = 2 giving
graph{ ((x^2+y^2)^0.51 +2(x + y))( (x^2+y^2)^0.51 +2(x + y)) = 0[2 2 2 2]}
Changing c to 1, in the above assignment, a parabola is traced.
graph{ (x^2+y^2)^0.51 +(x + y) = 0[2 2 2 2]}
Changing c to 0.5, in the above assignment, an ellipse is traced.
graph{ (x^2+y^2)^0.51 +0.5(x + y) = 0[4 2 4 2]}
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Here, crossmultiplying and rearranging,
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To convert ( r = \frac{6}{3  4\cos(\theta)} ) into Cartesian form, follow these steps:

Replace ( r ) with its Cartesian form equivalent using the conversion formulas: [ r = \sqrt{x^2 + y^2} ]

Rewrite the given polar equation with its Cartesian form equivalent: [ \sqrt{x^2 + y^2} = \frac{6}{3  4\cos(\theta)} ]

Square both sides of the equation to eliminate the square root: [ x^2 + y^2 = \frac{36}{(3  4\cos(\theta))^2} ]

Use the trigonometric identity ( \cos^2(\theta) + \sin^2(\theta) = 1 ) to express ( \cos(\theta) ) in terms of ( x ) and ( y ): [ \cos(\theta) = \frac{x}{r} ]

Substitute ( \frac{x}{r} ) for ( \cos(\theta) ) in the equation: [ x^2 + y^2 = \frac{36}{\left(3  4\frac{x}{r}\right)^2} ]

Substitute ( r = \sqrt{x^2 + y^2} ) into the equation: [ x^2 + y^2 = \frac{36}{\left(3  4\frac{x}{\sqrt{x^2 + y^2}}\right)^2} ]

Simplify the equation and express it in terms of ( x ) and ( y ) only.
By following these steps, you can convert the polar equation ( r = \frac{6}{3  4\cos(\theta)} ) into Cartesian form.
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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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