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Polar Coordinate Party
Transcript of Polar Coordinate Party
Quick review: Cartesian coordinates are the coordinates that we mostly deal with. They are our (x,y) coordinates.
The Relationship Between Cartesian and Polar Coordinates
Recap: Cartesian coordinates are (x,y)
Polar coordinates are (r, T).
is a curve whose equation is expressed through polar coordinates
Polar Coordinate Party
Polar coordinates are a coordinate system expressed in the form of (r, T). The r represents the length of the ray beginning from the origion (0,0).
The r is the length of the ray which begins from the pole (0,0).
The T is the angle between the ray and the polar axis (x-axis).
To convert from Cartesian to polar coordinates, we must find r and theta. We find r using the Pythagorean Theorem, and theta using tangent function:
ex. Find the polar coordinate of the cartesian coordinate (2,5).
Step 1: The first thing we need to do is find r.
r^2 = x^2 + y^2
r^2 = (2)^2 + (5)^2
r^2 = 4 + 25
so r is approximately 5.3852
Step 2: We now need to find theta.
tan(theta) = x/y
theta = inverse tan of (x/y)
theta = inverse tan of (2/5)
so theta is approximately 0.3804 radians
The equivalent polar coordinate is (5.3852, 0.3804).
Graphing a Basic Polar Curve on a Calculator:
Why are polar coordinates useful?
Polar coordinates are often used for navigation purposes (the angle and distance can help locate destinations).
The angles are measured clockwise with 90, 180, 270, and 360 corresponding to East, South, West, and North respectively.
These coordinates are also seen in physics when calculating different vector components.
They are often used in engineering
Though there aren't any calculations involved with radially symmetrical plants, polar coordinates can be visualized within them
The centre can be seen as the pole
They are also the base of the Archimedean spiral
example: P = (-3,10)
r =√((-3)^2 + 10^2) = √109 = 10.4 to 1 decimal place
P is in Quadrant II
theta = tan^-1(10/-3) = tan^-1(-3.33...)
The calculator value for tan-1(-3.33...) is -73.3°
***The rule for Quadrant II is: Add 180° to the calculator value
theta = -73.3° + 180° = 106.7°
therefore, the Polar Coordinates for the point (-3, 10) are (10.4, 106.7°)
Quadrant Value of tan^-1
I Use the calculator value
II Add 180° to the calculator value
III Add 180° to the calculator value
IV Add 360° to the calculator value
About the Value of Tan^-1