140k views
4 votes
Find the polar coordinates, 0 ≤ theta < 2π and r ≥ 0, of the following points given in Cartesian coordinates.

a.) (3√3, 3)
b.) (-3√3, 3)
c.) (-2, -2√3)

User Umlcat
by
8.5k points

1 Answer

5 votes

Answer:

(r; θ) =

  • (6; π/6)
  • (6; 5π/6)
  • (4; 4π/3)

Explanation:

You want the polar coordinates corresponding to the Cartesian coordinates ...

  • a.) (3√3, 3)
  • b.) (-3√3, 3)
  • c.) (-2, -2√3)

Coordinate conversion

The relation between polar and cartesian coordinates is ...

(x, y) ⇒ (√(x²+y²); arctan(y/x))

where the arctangent function takes quadrant into account.

Application

The attachment shows a calculator's output where the Cartesian coordinates are translated to the complex plane. The negative angle is converted to a positive angle by adding 2π radians.

As an example of how this works, we can use ...

c) (-2, -2√3) ⇒ (√((-2)² +(-2√3)²); arctan((-2√3)/-2))

⇒ (√16; arctan(√3)) . . . . . . where the angle is a 3rd quadrant angle

⇒ (4; π+π/3) = (4; 4π/3)

The polar coordinates are ...

a.) (6; π/6)

b.) (6; 5π/6)

c.) (4; 4π/3)

__

Additional comment

There are several possible notations for polar coordinates. We have used one that is similar to the notation (x, y) for Cartesian coordinates, but uses a semicolon (;) separator to identify the ordered pair as polar coordinates.

The calculator uses a notation r∠θ, which we like for its compactness. Some calculators write both forms as vectors [x, y] or [r, θ] and leave it to the user to interpret the values appropriately.

Other notations used for polar coordinates are r(cos(θ), sin(θ)) or r(cos(θ)+i·sin(θ)) or r cis θ. The last of these is an abbreviation of the one before.

<95141404393>

Find the polar coordinates, 0 ≤ theta < 2π and r ≥ 0, of the following points given-example-1
User DMEM
by
7.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories