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)
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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.
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