# 2,3 Polar and Cartesian Coordinates

<-- Enter Point

Given the point (3,4), perform or determine the following:
Ordered Pair Detail
Polar to Cartesian
Cartesian to Polar
Equivalent Coordinates
Rotate 90 Degrees
Rotate 180 Degrees
Rotate 270 Degrees
Reflect over origin
Reflect over y-axis
Reflect over x-axis
Evaluate the ordered pair (3,4)

We start at the coordinates (0,0)
Since our x coordinate of 3 is positive, we move up on the graph 3 space(s)
Since our y coordinate of 4 is positive, we move right on the graph 4 space(s)

Determine the Quadrant that (3,4) lies in
Since 3>0 and 4>0, (3,4) is in Quadrant I

Convert the point (3,4°) 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) = (3,4°)
(rcosθ,rsinθ) = (3cos(4),3sin(4))
(rcosθ,rsinθ) = (3(0.39),3(0.546))
(rcosθ,rsinθ) = (2.9927,0.2093)
Therefore, (3,4°) in polar coordinates equals (2.9927,0.2093) in Cartesian coordinates

Since 2.9927>0 and 0.2093>0, (2.9927,0.2093) is in Quadrant I

Convert the Cartesian point (3,4) to polar coordinates

Cartesian Coordinates are denoted as (x,y)
Polar Coordinates are denoted as (r,θ)
(x,y) = (3,4)

The transformation for r is denoted below:
r = ±√x2 + y2
r = ±√32 + 42
r = ±√9 + 16
r = ±√25
r = ±5

The transformation for θ is denoted below:
θ = tan-1(y/x)
θ = tan-1(4/3)
θ = tan-1(3)

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

 θdegrees = 0.61 * 180 π

 θdegrees = 029 π

θdegrees = 53.13°
Therefore, (3,4) in Cartesian coordinates equals (5,53.13°) in Polar coordinates

Since 3>0 and 4>0, (3,4) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (3,4°)

Show clockwise equivalent coordinates by adding 360°
(3,4° + 360°)
(3,364°)

(3,4° + 360°)
(3,724°)

(3,4° + 360°)
(3,1084°)

Method 2: Show equivalent coordinates by taking -(r) and adding 180°
(-1 * 3,4° + 180°)
(-3,184°)

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

Take (3, 4) and rotate 90 degrees denoted as R90°

The formula for rotating a point 90° is R90°(x, y) = (-y, x)
R90°(3, 4) = (-(4), 3)
R90°(3, 4) = (-4, 3)

Take (3, 4) and rotate 180 degrees denoted as R180°

The formula for rotating a point 180° is R180°(x, y) = (-x, -y)
R180°(3, 4) = (-(3), -(4))
R180°(3, 4) = (-3, -4)

Take (3, 4) and rotate 270 degrees denoted as R270°

The formula for rotating a point 270° is R270°(x, y) = (y, -x)
R270°(3, 4) = (4, -(3))
R270°(3, 4) = (4, -3)

Take (3, 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(3, 4) = (-(3), -(4))
rorigin(3, 4) = (-3, -4)

Take (3, 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(3, 4) = (-(3), 4)
ry-axis(3, 4) = (-3, 4)

Take (3, 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(3, 4) = (3, -(4))
rx-axis(3, 4) = (3, -4)