Polar and Cartesian Coordinates

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Given the point (1,4), 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 (1,4)
The abcissa is the absolute value of the x-coordinate, or perpendicular distance to the y-axis
Abcissa = |1| = 1

Determine the ordinate for (1,4)
The ordinate is the absolute value of the y-coordinate, or perpendicular distance to the x-axis
Ordinate = |4| = 4

Evaluate the ordered pair (1,4)

We start at the coordinates (0,0)
Since our x coordinate of 1 is positive, we move up on the graph 1 space(s)
Since our y coordinate of 4 is positive, we move right on the graph 4 space(s)

Determine the Quadrant that (1,4) lies in
Since 1>0 and 4>0, (1,4) is in Quadrant I

Convert the point (1,4°) 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) = (1,4°)
(rcosθ,rsinθ) = (1cos(4),1sin(4))
(rcosθ,rsinθ) = (1(0.39),1(0.546))
(rcosθ,rsinθ) = (0.9976,0.0698)
Therefore, (1,4°) in polar coordinates equals (0.9976,0.0698) in Cartesian coordinates

Since 0.9976>0 and 0.0698>0, (0.9976,0.0698) is in Quadrant I

Convert the Cartesian point (1,4) to polar coordinates

Cartesian Coordinates are denoted as (x,y)
Polar Coordinates are denoted as (r,θ)
(x,y) = (1,4)

The transformation for r is denoted below:
r = ±√x2 + y2
r = ±√12 + 42
r = ±√1 + 16
r = ±√17
r = ±4.7

The transformation for θ is denoted below:
θ = tan-1(y/x)
θ = tan-1(4/1)
θ = tan-1(4)
θradians =

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

θdegrees = * 180
π

θdegrees =025
π

θdegrees = 75.96°
Therefore, (1,4) in Cartesian coordinates equals (4.7,75.96°) in Polar coordinates

Since 1>0 and 4>0, (1,4) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (1,4°)

Show clockwise equivalent coordinates by adding 360°
(1,4° + 360°)
(1,364°)

(1,4° + 360°)
(1,724°)

(1,4° + 360°)
(1,1084°)

Method 2: Show equivalent coordinates by taking -(r) and adding 180°
(-1 * 1,4° + 180°)
(-1,184°)

Method 3: Show equivalent coordinates by taking -(r) and subtracting 180°
(-1 * 1,4° - 180°)
(-1,-176°)

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 = (-1, -4)

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 = (1, -4)

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 = (-1, 4)

Take (1, 4) and rotate 90 degrees denoted as R90°

The formula for rotating a point 90° is R90°(x, y) = (-y, x)
R90°(1, 4) = (-(4), 1)
R90°(1, 4) = (-4, 1)

Take (1, 4) and rotate 180 degrees denoted as R180°

The formula for rotating a point 180° is R180°(x, y) = (-x, -y)
R180°(1, 4) = (-(1), -(4))
R180°(1, 4) = (-1, -4)

Take (1, 4) and rotate 270 degrees denoted as R270°

The formula for rotating a point 270° is R270°(x, y) = (y, -x)
R270°(1, 4) = (4, -(1))
R270°(1, 4) = (4, -1)

Take (1, 4) 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(1, 4) = (-(1), -(4))
rorigin(1, 4) = (-1, -4)

Take (1, 4) 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(1, 4) = (-(1), 4)
ry-axis(1, 4) = (-1, 4)

Take (1, 4) 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(1, 4) = (1, -(4))
rx-axis(1, 4) = (1, -4)