The cartesian equation of $r equals a s i n space 2 theta$ is

${(x^{2} + y^{2})}^{3} = 4 a^{2} x^{2} y^{2}$
${(x^{2} - y^{2})}^{3} = 4 a^{2} x^{2} y^{2}$
${(x^{2} + y^{2})}^{3} = 2 a^{2} x^{2} y^{2}$
${(x^{2} - y^{2})}^{3} = 2 a^{2} x^{2} y^{2}$

Hint:

The term "polar coordinate system" refers to a two-dimensional coordinate system where each point's location on a plane is determined by its distance from a reference point and its angle with respect to a reference direction. Here we have to find the polar equation of $r equals a s i n space 2 theta$ .

The correct answer is: $open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent$

Using the formula, we may generate an endless number of polar coordinates for a single coordinate point.
(r, +2n) or (-r, +(2n+1)) are possible formulas, where n is an integer.
If measured in the opposite direction, the value of is positive.
When calculated counterclockwise, the value of is negative.
If laid off at the terminal side of, the value of r is positive.
If r is terminated at the prolongation through the origin from the terminal side of, it has a negative value.
The equation is given as
$r equals a sin space 2 theta N o w space w e space k n o w space t h a t colon sin space 2 theta equals 2 s i n theta cos theta A p p l y i n g space t h i s comma space w e space g e t colon r equals 2 a s i n theta cos theta M u l t i p l y i n g space b y space r squared space o n space b o t h space s i d e s comma space w e space g e t colon r cubed equals 2 a left parenthesis r s i n theta right parenthesis left parenthesis r cos theta right parenthesis P u t t i n g space r cos theta equals x space a n d space space r s i n theta equals y comma space w e space g e t colon r equals square root of x squared plus y squared end root left parenthesis x squared plus y squared right parenthesis to the power of 3 divided by 2 end exponent equals 2 a y x space S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon 4 a squared x squared y squared equals left parenthesis x squared plus y squared right parenthesis cubed$

Here we used the concept of the polar coordinate system and also the trigonometric ratios to find the solution. So the equation is $4 a squared x squared y squared equals left parenthesis x squared plus y squared right parenthesis cubed$ .

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The cartesian equation of $r equals a s i n space 2 theta$ is

${(x^{2} + y^{2})}^{3} = 4 a^{2} x^{2} y^{2}$
${(x^{2} - y^{2})}^{3} = 4 a^{2} x^{2} y^{2}$
${(x^{2} + y^{2})}^{3} = 2 a^{2} x^{2} y^{2}$
${(x^{2} - y^{2})}^{3} = 2 a^{2} x^{2} y^{2}$

The correct answer is: $open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent$

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The cartesian equation of is

The term "polar coordinate system" refers to a two-dimensional coordinate system where each point's location on a plane is determined by its distance from a reference point and its angle with respect to a reference direction. Here we have to find the polar equation of .

The correct answer is:

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The cartesian equation of $r equals a s i n space 2 theta$ is

The correct answer is: $open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent$