How do you convert the polar equation #r=3sintheta# into rectangular form?

Answer 1

#x^2+(y-3/2)^2=9/4#

#x=rcos(theta)#
#y=rsin(theta)#
#r^2=x^2+y^2#

Multiply the equation by #r#

#r*r=3rsin(theta)#

Simplify

#r^2=3rsin(theta)#

Make appropriate substitutions

#x^2+y^2=3y#

Gather all of the terms to the same side

#x^2+y^2-3y=0#

Complete square using the coefficient of #y# variable

#(-3/2)^2=9/4#

Add #9/4# to both sides the equation to keep it balanced. The constant #9/4# allows you make a perfect square trinomial.

#x^2+y^2-3y+9/4=9/4#

Rewrite

#x^2+(y-3/2)^2=9/4#

Check out a tutorial on converting an equation from polar to rectangular

Check out a tutorial on completing the square graphically

Check out a tutorial on completing the square analytically

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Answer 2

To convert the polar equation ( r = 3\sin(\theta) ) into rectangular form, you can use the relationships between polar and rectangular coordinates:

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

Substitute ( r = 3\sin(\theta) ) into these equations:

[ x = 3\sin(\theta)\cos(\theta) ] [ y = 3\sin^2(\theta) ]

Using the trigonometric identity ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) ), you can rewrite ( x ) in terms of ( \sin(2\theta) ):

[ x = \frac{3}{2}\sin(2\theta) ]

Thus, the rectangular form of the polar equation ( r = 3\sin(\theta) ) is ( x = \frac{3}{2}\sin(2\theta) ) and ( y = 3\sin^2(\theta) ).

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Answer from HIX Tutor

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.

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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