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Precalculus : Convert Rectangular Coordinates To Polar Coordinates and vice versa

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Example Questions

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:

$\small (10\sqrt3,10)$

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

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

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

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

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

Convert the polar coordinates to rectangular coordinates:

$(4, \pi)$

Possible Answers:

(0, -4)

(4, 0)

(-4, -4)

(-4, 0)

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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Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$\left(8, \frac{7\pi}{6}\right)$

Possible Answers:

$(-4\sqrt2, 4\sqrt2)$

$(4\sqrt2, 4\sqrt2)$

$\left(-\frac{8\sqrt3}{2}, -4\right)$

$(\sqrt2, \sqrt2)$

Correct answer:

$\left(-\frac{8\sqrt3}{2}, -4\right)$

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,

$x=8\cos \frac{7\pi}{6}=8\left(-\frac{\sqrt3}{2}\right)=-\frac{8\sqrt3}{2}$

$y=8 \sin \frac{7\pi}{6}=8\left(-\frac{1}{2}\right)=-4$

The rectangular coordinates are $\left(-\frac{8\sqrt3}{2}, -4\right)$

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

How could you express $\small \small \left(-5, \frac{\pi}{3}\right)$ in rectangular coordinates?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The polar coordinates given have an angle of $\small \frac{\pi}{3}$ but a negative radius, so our coordinates are located in quadrant III.

This means x and y are both negative. You can figure out these x and y coordinates using trigonometric ratios, or since the angle is $\small \frac{\pi}{3}$ , special right triangles.

The hypotenuse of this triangle is 5, but in the special right triangle it's 2, so we know we're multiplying each side by $\small \frac{5}{2}$ .

That makes the x-coordinate or adjacent side be

$\small \frac{5}{2}*1 = \frac{5}{2}$

and the y-coordinate or opposite side be

$\small \frac{5}{2}*\sqrt{3} = \frac{5\sqrt3}{2}$ .

In this case, once again, both are negative, so our answer is

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

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Example Question #6 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express $\left(10, \frac{\pi}{10}\right)$ in rectangular coordinates?

Round to the nearest hundredth.

Possible Answers:

(3.09, 9.51)

(9.51, 3.09)

(10, 0.05)

(3.25, 3.25)

(0.05, 10)

Correct answer:

(9.51, 3.09)

Explanation:

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

Polar to rectangular a

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

$cos\left(\frac{\pi}{10}\right) = \frac{x}{10}$

$0.951 \approx \frac{x}{10}$ multiply both sides by 10.

9.51 = x

We can figure out the y-coordinate by using the sine:

$sin\left(\frac{\pi}{10}\right) = \frac{y}{10}$

$0.309 \approx \frac{y}{10}$

3.09 = y

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Precalculus : Convert Rectangular Coordinates To Polar Coordinates and vice versa

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

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 #2 : 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

Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

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

Example Question #6 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

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