 # Ordered Pair

## Plot this on the Cartesian Graph: ## Determine the abcissa for (5,30)

Abcissa = absolute value of x-value
Perpendicular distance to the y-axis
Abcissa = |5| = 5

## Determine the ordinate for (5,30)

Ordinate = absolute value of y-value
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 for (5,30)

Since 5>0 and 30>0

Convert the point (5,30°) from
polar to Cartesian

## The formula for this is below:

Polar Coordinates are (r,θ)
Cartesian Coordinates are (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(0.86602540408359),5(0.49999999948186))
(rcosθ,rsinθ) = (4.3301,2.5)
(5,30°) = (4.3301,2.5)

## Determine the quadrant for (4.3301,2.5)

Since 4.3301>0 and 2.5>0

Convert (5,30) to polar

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

## Transform r:

r = ±√x2 + y2
r = ±√52 + 302
r = ±√25 + 900
r = ±√925
r = ±30.413812651491

θ = tan-1(y/x)
θ = tan-1(30/5)
θ = tan-1(6)

## Convert our angle to degrees

 Angle in Degrees  = Angle in Radians * 180 π

 θdegrees  = 1.4056476493803 * 180 π

 θdegrees  = 253.01657688845 π

θdegrees = 80.54°
Therefore, (5,30) = (30.413812651491,80.54°)

## Determine the quadrant for (5,30)

Since 5>0 and 30>0

(5,30° + 360°)
(5,390°)

(5,30° + 360°)
(5,750°)

(5,30° + 360°)
(5,1110°)

## Method 2: -(r) + 180°

(-1 * 5,30° + 180°)
(-5,210°)

## Method 3: -(r) - 180°

(-1 * 5,30° - 180°)
(-5,-150°)

## Determine symmetric point

If (x,y) is symmetric to the origin:
then the point (-x,-y) is also on the graph
(-5, -30)

## Determine symmetric point

If (x,y) is symmetric to the x-axis:
then the point (x, -y) is also on the graph
(5, -30)

## Determine symmetric point

If (x,y) is symmetric to the y-axis:
then the point (-x, y) is also on the graph
(-5, 30)

Take (5, 30) and rotate 90 degrees
We call this 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
We call this 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
We call this 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
We call this rorigin

Formula for reflecting 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
We call this ry-axis

Formula for reflecting 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
We call this rx-axis

Formula for reflecting over the x-axis is:
rx-axis(x, y) = (x, -y)
rx-axis(5, 30) = (5, -(30))
rx-axis(5, 30) = (5, -30)

### How does the Ordered Pair Calculator work?

This calculator handles the following conversions:
* Ordered Pair Evaluation and symmetric points including the abcissa and ordinate
* Polar coordinates of (r,θ°) to Cartesian coordinates of (x,y)
* Cartesian coordinates of (x,y) to Polar coordinates of (r,θ°)
* Quadrant (I,II,III,IV) for the point entered.
* Equivalent Coordinates of a polar coordinate
* Rotate point 90°, 180°, or 270°
* reflect point over the x-axis
* reflect point over the y-axis
* reflect point over the origin
This calculator has 1 input.

### What 2 formulas are used for the Ordered Pair Calculator?

1. Cartesian Coordinate = (x, y)
2. (r,θ) → (x,y) = (rcosθ,rsinθ)

For more math formulas, check out our Formula Dossier

### What 15 concepts are covered in the Ordered Pair Calculator?

cartesian
a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length
coordinates
A set of values that show an exact position
cos
cos(θ) is the ratio of the adjacent side of angle θ to the hypotenuse
degree
A unit of angle measurement, or a unit of temperature measurement
ordered pair
A pair of numbers signifying the location of a point
(x, y)
point
an exact location in the space, and has no length, width, or thickness
polar
a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction
1 of 4 sections on the Cartesian graph. Quadrant I: (x, y), Quadrant II (-x, y), Quadrant III, (-x, -y), Quadrant IV (x, -y)
1 of 4 sections on the Cartesian graph. Quadrant I: (x, y), Quadrant II (-x, y), Quadrant III, (-x, -y), Quadrant IV (x, -y)
rectangular
A 4-sided flat shape with straight sides where all interior angles are right angles (90°).
reflect
a flip creating a mirror image of the shape
rotate
a motion of a certain space that preserves at least one point.
sin
sin(θ) is the ratio of the opposite side of angle θ to the hypotenuse
x-axis
the horizontal plane in a Cartesian coordinate system
y-axis
the vertical plane in a Cartesian coordinate system