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For questions 1-5, the polar coordinates of a point are given. Find the rectangular coordinates of each point.

1. (5,(π / 4))

2. (-2,(π / 6))

3. (-1,(-2π / 3))

4. (3,225°)

5. (-3,330°)

Respuesta :

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QUESTION 1

We want to convert [tex](5,\frac{\pi}{4})[/tex] from polar coordinates to rectangular coordinates.

We use the formula;

[tex]x=r\cos \theta[/tex]

[tex]y=r\sin \theta[/tex]

We plug in [tex]r=5[/tex] and [tex]\theta=\frac{\pi}{4}[/tex] to obtain;

[tex]x=5\cos \frac{\pi}{4}=\frac{5\sqrt{2}}{2}[/tex]

[tex]y=5\sin \frac{\pi}{4}=\frac{5\sqrt{2}}{2}[/tex]

The rectangular coordinate is [tex](\frac{5\sqrt{2}}{2},\frac{5\sqrt{2}}{2})[/tex]

QUESTION 2

We want to convert [tex](-2,\frac{\pi}{6})[/tex] from polar coordinates to rectangular coordinates.

We use the formula;

[tex]x=r\cos \theta[/tex]

[tex]y=r\sin \theta[/tex]

We plug in [tex]r=-2[/tex] and [tex]\theta=\frac{\pi}{6}[/tex] to obtain;

[tex]x=-2\cos \frac{\pi}{6}=-1[/tex]

[tex]y=-2\sin \frac{\pi}{6}=-\sqrt{3}[/tex]

The rectangular coordinate is [tex](-1,-\sqrt{3})[/tex]

QUESTION 3

We want to convert [tex](-1,\frac{-2\pi}{3})[/tex] from polar coordinates to rectangular coordinates.

We use the formula;

[tex]x=r\cos \theta[/tex]

[tex]y=r\sin \theta[/tex]

We plug in [tex]r=-1[/tex] and [tex]\theta=\frac{-2\pi}{3}[/tex] to obtain;

[tex]x=-1\cos \frac{-2\pi}{3}=\frac{1}{2}[/tex]

[tex]y=-1\sin \frac{-2\pi}{3}=\frac{\sqrt{3}}{2}[/tex]

The rectangular coordinate is [tex](\frac{1}{2},\frac{\sqrt{3}}{2})[/tex]

QUESTION 4

We want to convert [tex](3,225\degree)[/tex] from polar coordinates to rectangular coordinates.

We use the formula;

[tex]x=r\cos \theta[/tex]

[tex]y=r\sin \theta[/tex]

We plug in [tex]r=3[/tex] and [tex]\theta=225\degree[/tex] to obtain;

[tex]x=3\cos (225\degree)=-3\frac{\sqrt{2}}{2}[/tex]

[tex]y=3\sin(225\degree)=-\frac{3\sqrt{2}}{2}[/tex]

The rectangular coordinate is [tex](-\frac{3\sqrt{2}}{2},-3\frac{\sqrt{2}}{2})[/tex]

QUESTION 5

We want to convert [tex](-3,330\degree)[/tex] from polar coordinates to rectangular coordinates.

We use the formula;

[tex]x=r\cos \theta[/tex]

[tex]y=r\sin \theta[/tex]

We plug in [tex]r=-3[/tex] and [tex]\theta=330\degree[/tex] to obtain;

[tex]x=-3\cos (330\degree)=-\frac{3\sqrt{3}}{2}[/tex]

[tex]y=-3\sin(330\degree)=\frac{3}{2}[/tex]

The rectangular coordinate is [tex](-\frac{3\sqrt{3}}{2},\frac{3}{2})[/tex]

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