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Precalculus : Polar Coordinates and Complex Numbers

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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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Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates $\small (10, 150^{\circ})$ to rectangular form.

Possible Answers:

$\small (-10,-10\sqrt3)$

$\small (-10\sqrt3,5)$

$\small (10\sqrt3,10)$

$\small (-5\sqrt3,5)$

$\small (5,-5\sqrt3)$

Correct answer:

$\small (-5\sqrt3,5)$

Explanation:

We begin by recalling that polar coordinates are expressed in the form $\small (r,\theta)$ , where $\small r$ is the radius (the distance from the origin to the point) and $\small \theta$ is the angle formed between the postive x-axis and the radius.

We can find our x-coordinate and y-coordinate in rectangular form quite easily by keeping in mind two equations.

$\small cos(\theta)=\frac{x}{r}$ or $\small x=r(cos(\theta))$

$\small sin(\theta)=\frac{y}{r}$ or $\small y=r(sin(\theta))$

Substituting in both of these gives respectively

$\small x=(10)cos(150^{\circ})=10(-\frac{\sqrt3}{2})=-5\sqrt3$

$\small y=10(sin(150^\circ))=10(\frac{1}{2})=5$

Therefore, the rectangular coordinates of our point are $\small (-5\sqrt3,5)$

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Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(5, 60)

Possible Answers:

(5, 9)

(4.5, 1.22)

(2.5, 4.33)

(1.56, 1.42)

Correct answer:

(2.5, 4.33)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=5\cos60=2.5$

y=5sin60=4.33

The rectangular coordinates are (2.5, 4.33)

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Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(8, 45)

Possible Answers:

(9.54, 9.33)

(2.33, 2.33)

(4.54, 2.36)

(5.66, 5.66)

Correct answer:

(5.66, 5.66)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=8\cos 45=5.66$

$y=8 \sin 45=5.66$

The rectangular coordinates are (5.66, 5.66)

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Example Question #4 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(10, 35)

Possible Answers:

(6.32, 7.12)

(6.35, 8.11)

(8.19, 5.74)

(8.11, 1.25)

Correct answer:

(8.19, 5.74)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=10\cos 35=8.19$

$y=10\sin 35=5.74$

The rectangular coordinates are (8.19, 5.74)

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Example Question #5 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$(4, \pi)$

Possible Answers:

(4, 0)

(-4, 0)

(-4, -4)

(0, -4)

Correct answer:

(-4, 0)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=4\cos \pi=4(-1)=-4$

$y=4 \sin \pi=4(0)=0$

The rectangular coordinates are (-4, 0)

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Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$\left(5, \frac{2\pi}{3}\right)$

Possible Answers:

$\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

$\left(\frac{2}{5}, \frac{\sqrt3}{2}\right)$

(2.33, 21.2)

$\left(\frac{\sqrt2}{4}, \frac{5}{\sqrt3}\right)$

Correct answer:

$\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=5\cos \frac{2\pi}{3}=5\left(-\frac{1}{2}\right)=-\frac{5}{2}$

$y=5 \sin \frac{2\pi}{3}=5\left(\frac{\sqrt3}{2}\right)=\frac{5\sqrt3}{2}$

The rectangular coordinates are $\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

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Example Question #7 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular form:

$\left(10, \frac{4\pi}{3}\right)$

Possible Answers:

$(-5, -5\sqrt3)$

$(5, 5\sqrt3)$

$(-5\sqrt3, -5)$

$(5\sqrt3, -5)$

Correct answer:

$(-5, -5\sqrt3)$

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=10\cos \frac{4\pi}{3}=10\left(-\frac{1}{2}\right)=-5$

$y=10 \sin \frac{4\pi}{3}=10\left(-\frac{\sqrt3}{2}\right)=-5\sqrt3$

The rectangular coordinates are $(-5, -5\sqrt3)$

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Precalculus : Polar Coordinates and Complex Numbers

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

Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #4 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #5 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #7 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

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