#### Transcript of Polar Coordinate Party

What are polar coordinates?

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).

Polar Curves

A

polar curve

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).

Ex. r=5sinTheta

r= cos4theta

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

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