Polar Sweep - free

 Polar Sweep

Polar Sweep reinforces relationships between rectangular coordinates, polar coordinates, and basic trigonometry with a dynamic educational interface. Users can sweep through the circle to see changing values and helpful diagrams of right triangles for common angles.

Radians - as an introduction to radians this app shows decimal values, common PI fractions, and degree equivalents.

For a solid understanding of cartesian plane quadrants and polar coordinates, spend some time playing with this app.

Version 1.1: Now with a keypad for direct entry of (x,y) or (r, angle) coordinates, this app becomes one of the best rectangular-to-polar converters available.

The Angle (Greek letter Theta) Slider and Arrow Keys

This central slider allows the user to explore the cartesian plane. The values of (x,y) for the end of the red line are shown dynamically as the angle changes with the slider. The horizontal axis for “x” is positive to the right, and the vertical axis for “y” is positive going up. Sweep through the quadrants to see the relationships in action. The arrow keys allow stepping through the common angles.

The r Slider

The “r” slider is located at the bottom left of the screen. When r = 1 (default), x is the cosine of the angle, and y is the sine of the angle. Polar Sweep effectively becomes a trig table. To make the numbers easier to think about, try r = 10. In each case, adjusting r does not change the look of the polar diagram since the angle is not changing; “x” and “y” grow or shrink proportionally as “r” is adjusted. The scales of the axes would change, but in this case the red line “r” always represents full scale.

Radians

Radians can be shown by toggling the blue button located at the bottom right of the screen. Radians are simply another way of making angular measurements. They are very powerful, since the radian value indicates the arc length of the swept angle (when r = 1, the full circumference of a circle is 2*PI. Half a circle (180 degrees) represents an arc length of PI. Half of that (90 degrees) represents an arc length of PI/2). These ratios are true for any value of “r”, since the radius and circumference are proportional. Polar Sweep can help students get more comfortable with radians.

Standard Angles

Understanding a few standard angles will give reference points to understand the entire plane (cartesian plane -- a slice of 3-D space). Using the arrow keys will step through some common angles, all based on only two right triangles.

Direct Entry

Touch the upper gray blocks to enter (x, y) directly. Touch the red blocks to enter (r, angle). Refer to the images below for more.

Angles larger than 360 degrees ( or larger than 2*PI radians)

If a large angle is entered, it will be reduced. For example, if 380 degrees is entered, the resultant angle shown will be 20 degrees (one full revolution + 20 degrees). If -380 degrees is entered, the resultant angle shown will be 340 degrees (one full revolution clockwise + an additional 20 degrees clockwise, finishing 20 degrees below the x axis).

Polar Sweep launch image

(x,y) gray blocks for Rectangular Coordinates

(r, angle theta) red blocks for Polar Coordinates

Polar Sweep Layout

The central slider allows sweeping through a circle. The blue arrow keys advance through standard angles.

A slider for r (1 to 20) and the Degrees/Radians toggle are at the bottom of the screen.

(x, y) entry

Touch one of the gray blocks to bring up the keypad.

The x block is ready for input.

Enter the first x value (e.g. 8.25)

Touch the y block or the large green arrow button.

The y block is now ready for input.

Enter the y value (e.g. 4.125)

To complete, press the large green arrow button.

After pressing the large green arrow button, this display returns.

The red radius line in the diagram is 9.22 long at 26.565 degrees from the x axis. Its x component stretches 8.25 in the x direction, and its y component stretches up 4.125 in the y direction.

Pressing the Degrees/Radians key converts the main angle display to radians (or back to degrees).

In this case, the angle swept from the x axis to the red line is 0.463648 radians. The arc length of the sweep would be this times the radius (this is easier to compute if r = 1 or r = 10).

Right Triangle Formulas

The “info” button reveals this formula sheet.

The general length and angle definitions are at the top, and then the specifics for two standard triangles are illustrated.