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Calculus 2 : Polar

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Example Question #17 : Polar Calculations

Convert the cartesian point (1,1) into polar coordinates.

Possible Answers:

$(-8, \pi)$

(-1,1)

$(\sqrt3/2, \pi/6)$

$(\sqrt2, \pi/4)$

Correct answer:

$(\sqrt2, \pi/4)$

Explanation:

Cartesian coordinates are written in the form (x,y) .

In this problem, x=1 and y=1 .

Using the conversion formulas $r=\sqrt{x^2+y^2}$ and $\theta=tan^{-1}\left ( y/x\right )$ ,

$r=\sqrt{x^2+y^2}=\sqrt{1^2+1^2}=\sqrt2$

$\theta=tan^{-1}\left ( y/x\right )=tan^{-1}\left ( 1\right )=\pi/4$

The polar point is $(\sqrt2, \pi/4)$ .

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Example Question #18 : Polar Calculations

Convert the cartesian point (-3,4) into polar coordinates.

Possible Answers:

(-3,4)

$(\sqrt3/2, \pi/6)$

(5,5.356)

(-4,2.584)

Correct answer:

(5,5.356)

Explanation:

Cartesian coordinates are written in the form (x,y) .

In this problem, x=-3 and y=4 .

Using the conversion formulas $r=\sqrt{x^2+y^2}$ and $\theta=tan^{-1}\left ( y/x\right )$ ,

$r=\sqrt{x^2+y^2}=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

$\theta=tan^{-1}\left ( y/x\right )=tan^{-1}\left ( -4/3\right )=-0.927=5.356$

The polar point is (5,5.356) .

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Example Question #19 : Polar Calculations

Convert the cartesian point (5, 12) into polar coordinates.

Possible Answers:

(12,3.746)

(5, 12)

(13,1.176)

(-3,4)

Correct answer:

(13,1.176)

Explanation:

Cartesian coordinates are written in the form (x,y) .

In this problem, x=5 and y=12 .

Using the conversion formulas $r=\sqrt{x^2+y^2}$ and $\theta=tan^{-1}\left ( y/x\right )$ ,

$r=\sqrt{x^2+y^2}=\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$

$\theta=tan^{-1}\left ( y/x\right )=tan^{-1}\left ( 12/5\right )=1.176$

The polar point is (13,1.176) .

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Example Question #20 : Polar Calculations

Convert xy+7x^2=1+y into polar coordinates.

Possible Answers:

$r^2\left (sin\theta cos\theta+7sin^2\theta \right )-rcos\theta=7$

$rsin\theta=4$

$r^2\left (sin\theta cos\theta+7cos^2\theta \right )-rsin\theta=1$

$r^2\left (sin\theta cos\theta+7cos^2\theta \right )=-1$

Correct answer:

$r^2\left (sin\theta cos\theta+7cos^2\theta \right )-rsin\theta=1$

Explanation:

Substituting the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ into the cartesian equation,

xy+7x^2=1+y

we get

$(rcos\theta)(rsin\theta)+7(rcos\theta)^2=1+(rsin\theta)$

$r^2sin\theta cos\theta+7r^2cos^2\theta=1+rsin\theta$

$r^2\left (sin\theta cos\theta+7cos^2\theta \right )-rsin\theta=1$

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Example Question #21 : Polar Calculations

Convert 3x^2+3y^2=-4x into polar coordinates.

Possible Answers:

r=3

$r=-4cos^2\theta$

$r=\left (\frac{-4}{3} \right )cos\theta$

$r=\left (\frac{-4}{3} \right )sin\theta$

Correct answer:

$r=\left (\frac{-4}{3} \right )cos\theta$

Explanation:

Substituting the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ into the cartesian equation,

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

we get

$3(rcos\theta)^2+3(rsin\theta)^2=-4rcos\theta$

$3r^2cos^2\theta+3r^2sin^2\theta=-4rcos\theta$

$3r^2(cos^2\theta+sin^2\theta)=-4rcos\theta$

Using the identity, $cos^2\theta+sin^2\theta=1$

$3r^2=-4rcos\theta$

$r=\left (\frac{-4}{3} \right )cos\theta$

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Example Question #22 : Polar Calculations

Convert x^2+y^2=25 into polar coordinates.

Possible Answers:

$r=5cos\theta$

$r=5sin\theta$

$r=5sin\theta cos\theta$

r=5

Correct answer:

r=5

Explanation:

Substituting the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ into the cartesian equation,

x^2+y^2=25

we get

$(rcos\theta)^2+(rsin\theta)^2=25$

$r^2cos^2\theta+r^2sin^2\theta=25$

$r^2(cos^2\theta+sin^2\theta)=25$

Using the identity, $cos^2\theta+sin^2\theta=1$

r^2=25

$r=\sqrt{25}=5$

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Example Question #23 : Polar Calculations

Convert $r=4cos\theta$ into cartesian coordinates.

Possible Answers:

x^2+y^2=4x

x^2+4y^2=4x

x^2+y^2=4y

x^2+y^2=4x+4y

Correct answer:

x^2+y^2=4x

Explanation:

To make converting this expression easier, we will multiply both sides of the equation by , giving us

$r^2=4rcos\theta$

Substituting the conversion formulas r^2=x^2+y^2 and $x=rcos\theta$ into the polar equation, we get

x^2+y^2=4x

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Example Question #171 : Polar

Convert $r=-7sin\theta$ into cartesian coordinates.

Possible Answers:

x^2+y^2=x-y

7x^2+7y^2=-y

x^2+y^2=-7x

x^2+y^2=-7y

Correct answer:

x^2+y^2=-7y

Explanation:

To make converting this expression easier, we will multiply both sides of the equation by , giving us

$r^2=-7rsin\theta$

Substituting the conversion formulas r^2=x^2+y^2 and $y=rsin\theta$ into the polar equation, we get

x^2+y^2=-7y

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Example Question #25 : Polar Calculations

Convert $r=6cos\theta+2sin\theta$ into cartesian coordinates.

Possible Answers:

6x^2+y^2=x+2y

6x^2+2y^2=x-y

x^2+y^2=6x+2y

6x^2+2y^2=6x+2y

Correct answer:

x^2+y^2=6x+2y

Explanation:

To make converting this expression easier, we will multiply both sides of the equation by , giving us

$r^2=r(6cos\theta+2sin\theta)=6rcos\theta+2rsin\theta$

Substituting the conversion formulas r^2=x^2+y^2 , $x=rcos\theta$ and $y=rsin\theta$ into the polar equation, we get

x^2+y^2=6x+2y

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Example Question #21 : Polar Calculations

Convert $\large \large \large (1,-1)$ into polar coordinates.

Possible Answers:

$\large \left ( -2, \frac {\pi}{4}\right )$

$\large \left ( -2, -\frac {\pi}{4}\right )$

$\large \left ( 2, \frac {\pi}{4}\right )$

$\large \left ( 2, -\frac {\pi}{4}\right )$

Correct answer:

$\large \left ( 2, -\frac {\pi}{4}\right )$

Explanation:

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Calculus 2 : Polar

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #17 : Polar Calculations

Example Question #18 : Polar Calculations

Example Question #19 : Polar Calculations

Example Question #20 : Polar Calculations

Example Question #21 : Polar Calculations

Example Question #22 : Polar Calculations

Example Question #23 : Polar Calculations

Example Question #171 : Polar

Example Question #25 : Polar Calculations

Example Question #21 : Polar Calculations

All Calculus 2 Resources

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