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Calculus 2 : Parametric, Polar, and Vector

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #161 : Polar

Convert the polar point $(-5, 5 \pi/6)$ into cartesian coordinates.

Possible Answers:

$(-5, 5 \pi/6)$

$( 5\sqrt3/4,5\sqrt3/4)$

$(5\sqrt3/2, -5/2)$

$(5/2, 5\sqrt3/2)$

Correct answer:

$(5\sqrt3/2, -5/2)$

Explanation:

Cartesian coordinates are written in the form $(r,\theta)$ .

In this problem, r=-5 and $\theta=5 \pi/6$ .

Using the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ ,

$x=-5*cos(5\pi/6)=-5(-\sqrt3/2)=5\sqrt3/2$

$y=-5*sin(5\pi/6)=-5(1/2)=-5/2$

The cartesian point is $(5\sqrt3/2, -5/2)$ .

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Example Question #162 : Polar

Convert the polar point $(2, \pi/3)$ into cartesian coordinates.

Possible Answers:

$(\sqrt3,1)$

(1,1)

$(1,\sqrt3)$

$( 5\sqrt3/4,5\sqrt3/4)$

Correct answer:

$(1,\sqrt3)$

Explanation:

Cartesian coordinates are written in the form $(r,\theta)$ .

In this problem, r=2 and $\theta= \pi/3$ .

Using the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ ,

$x=2*cos(\pi/3)=2(1/2)=1$

$y=2*sin(\pi/3)=2(\sqrt3/2)=\sqrt3$

The cartesian point is $(1,\sqrt3)$ .

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Example Question #841 : Calculus Ii

Convert the polar point $(-8, \pi)$ into cartesian coordinates.

Possible Answers:

$(-8, \pi)$

(8,0)

(8,1)

$(1,\sqrt3)$

Correct answer:

(8,0)

Explanation:

Cartesian coordinates are written in the form $(r,\theta)$ .

In this problem, r=-8 and $\theta=\pi$ .

Using the conversion formulas $x=rcos\theta$ and $y=rsin\theta$ ,

$x=-8*cos(\pi)=-8(-1)=8$

$y=-8*sin(\pi)=-8(0)=0$

The cartesian point is (8,0) .

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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 #331 : Parametric, Polar, And Vector

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

Possible Answers:

(5,5.356)

(-3,4)

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

(-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 #171 : Polar

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

Possible Answers:

r=3

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

$r=-4cos^2\theta$

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

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 #172 : Polar

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

Possible Answers:

$r=5cos\theta$

$r=5sin\theta cos\theta$

$r=5sin\theta$

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 #171 : Polar

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

Possible Answers:

x^2+y^2=4x

x^2+4y^2=4x

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

x^2+y^2=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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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #161 : Polar

Example Question #162 : Polar

Example Question #841 : Calculus Ii

Example Question #17 : Polar Calculations

Example Question #331 : Parametric, Polar, And Vector

Example Question #19 : Polar Calculations

Example Question #20 : Polar Calculations

Example Question #171 : Polar

Example Question #172 : Polar

Example Question #171 : Polar

All Calculus 2 Resources

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