Question

How do you find the polar coordinates given (-2, 2)?

Answer 1

Hint: As the given coordinates are cartesian i.e., (x, y), hence we need it to convert to polar coordinates by applying respective formulas to convert. When each point on a plane of a two-dimensional coordinate system is decided by a distance from a reference point and an angle is taken from a reference direction, it is known as the polar coordinate system.

Formula used:
\[r = \sqrt {{x^2} + {y^2}} \]
r is the distance from the origin
x and y are the cartesian coordinates
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) + \pi \]
\[\theta \] is the angle of the coordinates.

Complete step-by-step solution:
To find the polar coordinates, let us consider the given cartesian coordinates as given (x, y) = (-2, 2), we need to convert these coordinates to polar coordinates.
To convert into polar coordinate, we need to find the distance from the origin to (x, y), hence the Radial component ‘r’ is given as
\[\Rightarrow r = \sqrt {{x^2} + {y^2}} \]
Substitute the given values of x and y in the formula
\[\Rightarrow r = \sqrt {{{\left( { - 2} \right)}^2} + {2^2}} \]
\[\Rightarrow r = \sqrt 8 \]
Therefore, the value of r is
\[\Rightarrow r = 2\sqrt 2 \]
As the x component is negative and y component is positive so we add \[\pi \] to the inverse tangent function, to place the resulting angle in the second quadrant as
\[\Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) + \pi \]
Substitute the given values of x and y we get
\[\Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{2}{{ - 2}}} \right) + \pi \]
\[\Rightarrow \theta = - \dfrac{\pi }{4} + \pi \]
\[\Rightarrow \theta = \dfrac{{3\pi }}{4}\]
Therefore, the polar coordinates are at
\[\Rightarrow \left( {2\sqrt {2,} \dfrac{{3\pi }}{4}} \right)\]

Thus the polar coordinates are \[\left( {2\sqrt {2,} \dfrac{{3\pi }}{4}} \right)\].

Additional information:
We can write an infinite number of polar coordinates for one coordinate point, using the formula
\[\left( {r,\theta + 2\pi n} \right)\], where n is an integer.

Note: The key point to find the polar coordinates is that we need to convert the given cartesian coordinates and in the Cartesian coordinate system, the distance of a point from the y-axis is called its x-coordinate and the distance of a point from the x-axis is called its y-coordinate.