Polar and Cartesian Coordinates

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Given the point (2,3), perform or determine the following:
Abcissa
Ordinate
Ordered Pair Detail
Quadrant
Polar to Cartesian
Cartesian to Polar
Equivalent Coordinates
Symmetric Points about the origin
Symmetric Points about the x-axis
Symmetric Points about the y-axis
Rotate 90 Degrees
Rotate 180 Degrees
Rotate 270 Degrees
Reflect over origin
Reflect over y-axis
Reflect over x-axis
Determine the abcissa for (2,3)
The abcissa is the absolute value of the x-coordinate, or perpendicular distance to the y-axis
Abcissa = |2| = 2

Determine the ordinate for (2,3)
The ordinate is the absolute value of the y-coordinate, or perpendicular distance to the x-axis
Ordinate = |3| = 3

Evaluate the ordered pair (2,3)

We start at the coordinates (0,0)
Since our x coordinate of 2 is positive, we move up on the graph 2 space(s)
Since our y coordinate of 3 is positive, we move right on the graph 3 space(s)

Determine the Quadrant that (2,3) lies in
Since 2>0 and 3>0, (2,3) is in Quadrant I

Convert the point (2,3°) from
polar coordinates to Cartesian (rectangular) coordinates

The formula for this is below:
Polar Coordinates are denoted as (r,θ)
Cartesian Coordinates are denoted as (x,y)
Polar to Cartesian Transformation is (r,θ) → (x,y) = (rcosθ,rsinθ)
(r,θ) = (2,3°)
(rcosθ,rsinθ) = (2cos(3),2sin(3))
(rcosθ,rsinθ) = (2(0.71),2(0.196))
(rcosθ,rsinθ) = (1.9973,0.1047)
Therefore, (2,3°) in polar coordinates equals (1.9973,0.1047) in Cartesian coordinates

Since 1.9973>0 and 0.1047>0, (1.9973,0.1047) is in Quadrant I

Convert the Cartesian point (2,3) to polar coordinates

Cartesian Coordinates are denoted as (x,y)
Polar Coordinates are denoted as (r,θ)
(x,y) = (2,3)

The transformation for r is denoted below:
r = ±√x2 + y2
r = ±√22 + 32
r = ±√4 + 9
r = ±√13
r = ±3.

The transformation for θ is denoted below:
θ = tan-1(y/x)
θ = tan-1(3/2)
θ = tan-1(1.5)
θradians = 0.33

Convert our angle to degrees from radians
Angle in Degrees =Angle in Radians * 180
π

θdegrees =0.33 * 180
π

θdegrees =452
π

θdegrees = 56.31°
Therefore, (2,3) in Cartesian coordinates equals (3.,56.31°) in Polar coordinates

Since 2>0 and 3>0, (2,3) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (2,3°)

Show clockwise equivalent coordinates by adding 360°
(2,3° + 360°)
(2,363°)

(2,3° + 360°)
(2,723°)

(2,3° + 360°)
(2,1083°)

Method 2: Show equivalent coordinates by taking -(r) and adding 180°
(-1 * 2,3° + 180°)
(-2,183°)

Method 3: Show equivalent coordinates by taking -(r) and subtracting 180°
(-1 * 2,3° - 180°)
(-2,-177°)

Determine symmetric point with respect to the origin
If the graph containing the point (x,y) is symmetric to the origin, then the point (-x,-y) is also on the graph
Therefore, our symmetric point with respect to the origin = (-2, -3)

Determine symmetric point with respect to the x-axis
If the graph containing the point (x,y) is symmetric to the x-axis, then the point (x, -y) is also on the graph
Therefore, our symmetric point with respect to the x-axis = (2, -3)

Determine symmetric point with respect to the y-axis
If the graph containing the point (x,y) is symmetric to the y-axis, then the point (-x, y) is also on the graph
Therefore, our symmetric point with respect to the y-axis = (-2, 3)

Take (2, 3) and rotate 90 degrees denoted as R90°

The formula for rotating a point 90° is R90°(x, y) = (-y, x)
R90°(2, 3) = (-(3), 2)
R90°(2, 3) = (-3, 2)

Take (2, 3) and rotate 180 degrees denoted as R180°

The formula for rotating a point 180° is R180°(x, y) = (-x, -y)
R180°(2, 3) = (-(2), -(3))
R180°(2, 3) = (-2, -3)

Take (2, 3) and rotate 270 degrees denoted as R270°

The formula for rotating a point 270° is R270°(x, y) = (y, -x)
R270°(2, 3) = (3, -(2))
R270°(2, 3) = (3, -2)

Take (2, 3) and reflect over the origin axis denoted as rorigin

The formula for reflecting a point over the origin is rorigin(x, y) = (-x, -y)
rorigin(2, 3) = (-(2), -(3))
rorigin(2, 3) = (-2, -3)

Take (2, 3) and reflect over the y-axis axis denoted as ry-axis

The formula for reflecting a point over the y-axis is ry-axis(x, y) = (-x, y)
ry-axis(2, 3) = (-(2), 3)
ry-axis(2, 3) = (-2, 3)

Take (2, 3) and reflect over the x-axis axis denoted as rx-axis

The formula for reflecting a point over the x-axis is rx-axis(x, y) = (x, -y)
rx-axis(2, 3) = (2, -(3))
rx-axis(2, 3) = (2, -3)