Polar and Cartesian Coordinates

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Given the point (5,30), 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 (5,30)
The abcissa is the absolute value of the x-coordinate, or perpendicular distance to the y-axis
Abcissa = |5| = 5

Determine the ordinate for (5,30)
The ordinate is the absolute value of the y-coordinate, or perpendicular distance to the x-axis
Ordinate = |30| = 30

Evaluate the ordered pair (5,30)

We start at the coordinates (0,0)
Since our x coordinate of 5 is positive, we move up on the graph 5 space(s)
Since our y coordinate of 30 is positive, we move right on the graph 30 space(s)

Determine the Quadrant that (5,30) lies in
Since 5>0 and 30>0, (5,30) is in Quadrant I

Convert the point (5,30°) 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) = (5,30°)
(rcosθ,rsinθ) = (5cos(30),5sin(30))
(rcosθ,rsinθ) = (5(0.59),5(0.86))
(rcosθ,rsinθ) = (4.3301,2.5)
Therefore, (5,30°) in polar coordinates equals (4.3301,2.5) in Cartesian coordinates

Since 4.3301>0 and 2.5>0, (4.3301,2.5) is in Quadrant I

Convert the Cartesian point (5,30) to polar coordinates

Cartesian Coordinates are denoted as (x,y)
Polar Coordinates are denoted as (r,θ)
(x,y) = (5,30)

The transformation for r is denoted below:
r = ±√x2 + y2
r = ±√52 + 302
r = ±√25 + 900
r = ±√925
r = ±30.

The transformation for θ is denoted below:
θ = tan-1(y/x)
θ = tan-1(30/5)
θ = tan-1(6)
θradians = 3

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

θdegrees =3 * 180
π

θdegrees =845
π

θdegrees = 80.54°
Therefore, (5,30) in Cartesian coordinates equals (30.,80.54°) in Polar coordinates

Since 5>0 and 30>0, (5,30) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (5,30°)

Show clockwise equivalent coordinates by adding 360°
(5,30° + 360°)
(5,390°)

(5,30° + 360°)
(5,750°)

(5,30° + 360°)
(5,1110°)

Method 2: Show equivalent coordinates by taking -(r) and adding 180°
(-1 * 5,30° + 180°)
(-5,210°)

Method 3: Show equivalent coordinates by taking -(r) and subtracting 180°
(-1 * 5,30° - 180°)
(-5,-150°)

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 = (-5, -30)

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 = (5, -30)

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 = (-5, 30)

Take (5, 30) and rotate 90 degrees denoted as R90°

The formula for rotating a point 90° is R90°(x, y) = (-y, x)
R90°(5, 30) = (-(30), 5)
R90°(5, 30) = (-30, 5)

Take (5, 30) and rotate 180 degrees denoted as R180°

The formula for rotating a point 180° is R180°(x, y) = (-x, -y)
R180°(5, 30) = (-(5), -(30))
R180°(5, 30) = (-5, -30)

Take (5, 30) and rotate 270 degrees denoted as R270°

The formula for rotating a point 270° is R270°(x, y) = (y, -x)
R270°(5, 30) = (30, -(5))
R270°(5, 30) = (30, -5)

Take (5, 30) 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(5, 30) = (-(5), -(30))
rorigin(5, 30) = (-5, -30)

Take (5, 30) 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(5, 30) = (-(5), 30)
ry-axis(5, 30) = (-5, 30)

Take (5, 30) 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(5, 30) = (5, -(30))
rx-axis(5, 30) = (5, -30)