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Precalculus : Convert Polar Equations To Rectangular Form and vice versa

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Example Questions

Example Question #51 : Polar Coordinates

Write the equation (x^2 -3y +y^2 ) ^2 = 4x^2 + 4y^2 in polar form

Possible Answers:

$r = 2+3 \sin \theta$

$r = 4 + 3 \sin \theta$

$r = 3 \sin \theta$

$r = 2 - 3 \sin \theta$

$r = 4 - 3 \sin \theta$

Correct answer:

$r = 2+3 \sin \theta$

Explanation:

First re-arrange the original equation so that the 4 is factored out on the right side, and put x^2 and y^2 next to each other:

(x^2 + y^2 - 3y)^2 = 4(x^2 + y^2)

Make the substitutions r^2 = x^2 + y^2 and $y = r \sin \theta$ :

$(r^2 - 3 r \sin \theta )^2 = 4r^2$ take the square root of both sides

$r^2 - 3 r \sin \theta = 2 r$ divide both sides by r

$r - 3 \cos \theta = 2$ add $3 \cos \theta$ to both sides

$r = 2 + 3 \cos \theta$

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Example Question #52 : Polar Coordinates

Which is equivalent to $y = \frac{1}{2}x^2$ ?

Possible Answers:

$r = \frac{2 \sin \theta }{\cos ^2 \theta}$

$r = \frac{\sin \theta}{ 2 \cos ^ 2 \theta }$

$r = \frac{2 \cos \theta }{\sin ^2 \theta}$

$r = \frac{\cos \theta }{ 2\sin ^2 \theta }$

$r = \frac{\cos ^2 \theta}{ 2 \sin \theta }$

Correct answer:

$r = \frac{2 \sin \theta }{\cos ^2 \theta}$

Explanation:

To convert, substitute $y = r \sin \theta$ and $x = r \cos \theta$ :

$r \sin \theta = \frac{ 1}{2} r^2 \cos ^2 \theta$ divide both sides by r

$\sin \theta = \frac{1}{2} r \cos ^2 \theta$ multiply both sides by 2

$2 \sin \theta = r \cos ^2 \theta$ divide both sides by cosine squared

$\frac{ 2\sin \theta }{ \cos ^2 \theta } = r$

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Example Question #53 : Polar Coordinates

Convert to polar form: y = 2x + 1

Possible Answers:

$r = \frac{ 2}{ \sin \theta - \cos \theta }$

$r = \frac{1}{ \sin \theta - 2 \cos \theta }$

$r = \sin \theta - 2 \cos \theta$

$r = \frac{ 1 }{ 2\sin \theta - \cos \theta }$

Correct answer:

$r = \frac{1}{ \sin \theta - 2 \cos \theta }$

Explanation:

Convert by making the substitutions $x = r \cos \theta$ and $y = r \sin \theta$ :

$r \sin \theta = 2 r \cos \theta + 1$ subtract $2 r \cos \theta$ from both sides

$r \sin \theta - 2 r \cos \theta = 1$ factor out r

$r (\sin \theta - 2 \cos \theta ) = 1$ divide

$r = \frac{ 1 }{ \sin \theta - 2 \cos \theta}$

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Precalculus : Convert Polar Equations To Rectangular Form and vice versa

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #51 : Polar Coordinates

Example Question #52 : Polar Coordinates

Example Question #53 : Polar Coordinates

All Precalculus Resources

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