Polar Coordinates! (Intro)
[color=#000000]The applet below illustrates how to plot points whose coordinates are written in [b]POLAR FORM[/b]. [br]You can adjust the values of [/color][color=#0000ff][b]r[/b][/color][color=#000000] and [/color][color=#980000][b]theta[/b][/color][color=#000000] by using the sliders or the input boxes. [br](This grapher will only accept [/color][color=#980000][b]theta[/b][/color][color=#000000] values ranging from [/color][color=#980000][b]-2pi to 2pi[/b][/color][color=#000000]). [br][br]Slide the [b]black slider[/b] to illustrate how polar coordinates are plotted. [br][/color][br][color=#000000]Interact with this applet for a few minutes, then answer the questions that follow. [br][/color]
Plot the following points using the calculator above: (3, pi/2) and (-3, 3pi/2). What do you notice?
They're coordinates for the same location (point)!
Can you come up with a different set of polar coordinates for the location given in the first question? If so, list it/them!
Possible solutions: (-3, -pi/2), (3, -3pi/2), and practically any point written in the form (3, pi/2 +2k*pi) or (-3, 3pi/2 + 2k*pi) or (-3, -pi/2 + 2k*pi) where k = any integer. [br][br][b]POLAR COORDINATE representations of points in the coordinate plane are NOT UNIQUE! [/b] There are infinitely many ways to express a location using polar coordinates. [b]Yet, there is only 1 way to express a point's location using RECTANGULAR COORDINATES (x, y)! [/b]
Quick (Silent) Demo
Derivatives in Polar Coordintes
[color=#ff0000]This applet was designed to serve as a "check", so to speak, for you when finding the derivative, dy/dx, of a [/color][color=#6aa84f]polar function[/color][color=#ff0000] at a certain point (t, r(t)). [/color][br][br][color=#0000ff][b]Note: [br]To move the tangent line along the function's graph, simply drag the t-slider provided in the upper left hand corner.[/b][/color][br][br][color=#9900ff][b]Another Note: [br]When inputting [i]t[/i] values expressed in terms of pi, simply type "pi" for pi. For example, to input 2pi/3, type "2pi/3". The fraction textbox will display your result. (The highest t-value you can input is 2pi.) [br][/b][/color][br][color=#980000][b]One more thing: [br]The fraction text (containing the pi factor) is only accurate if you use the input box to input a value written as a "ratio*pi". Otherwise, this value is approximate (just like the displayed decimal approximation.) [/b][/color]