![]() The output of the activities in Shodor's Interactivate are createdĭynamically by computer languages such as JavaScript. You may also try using the help feature of your browser. The links below provide instructions for enabling JavaScript dependent on your browser.Īfter enabling JavaScript, refresh the page. ![]() We have detected JavaScript as being disabled in your browser. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. The applet is similar to GraphIt, but instead allows users to explore the representation of a function in the polar coordinate system. ![]() But this is the general equation for a line in polar coordinates.Polar Coordinates: This activity allows the user to explore the polar coordinate system. So you really need to know the polar coordinates of that closest point in order to come up with this equation. So here the perameters of the lines are d, the distance between this point and the origin right? The closest point and beta, the angle between the positive axis and this line segment drawn to the closest point. And I get r cosine theta minus beta equals d and therefore r equals d over cosine of theta minus beta. Cosine of this angle theta minus beta equals side adjacent over hypotenuse and this is the side adjacent, this is the hypotenuse. And so the angle between the two between the 2 segments is going to be theta minus beta. And likewise, since the coordinates of this point are d beta, this length is d and this angle is beta. First of all let me observe that since the coordinates of point p here, this is just any arbitrary point on the line. Point of the origin would be point o and I can use right triangle trigonometry to get me an equation for this line. And we need that because it forms a right triangle here. And the point that's closest to the origin, if you drew a line from that point to the origin, it will be perpendicular to the given line. In order to do that I need to know the coordinates of the point that's closest to the origin. I want to find the equation of this line. Now, what about other lines? What about lines that don't pass through the origin? So draw a picture here of a line that doesn't pass through the origin. Now if you want something you know, something that's a little more numbery you can use your calculator to get inverse tangent of 5 and it's approximately 1.37. This is going to be the equation of the line y=5x in our polar coordinates. so this this number is a constant and so we have theta equals inverse tangent of 5. I could write theta equals inverse tangent of 5 though and that's a perfectly good equation. Well, unfortunately 5 is is not a real nice number. What about this one? Here we have y over x equals 5 and therefore tangent theta equals 5. So theta equals negative pi over 3, is the same as the line y equals negative root 3x. So what angle has a tangent of negative root 3? Well, it's negative pi over 3. So, what we have to do is figure out what theta equals. Now the first thing I would do is write this in the form y over x equals negative root 3 and then make the observation that y over x is the same as tangent theta. So let's use that and find the equations for some lines that pass through the origin like y equals negative root 3x. Now equations of this type theta equals k are all lines that pass through the origin and conversely all lines through the origin have equation of this form. Notice that every point has the same second coordinate pi over 6, that's the angle the angle from the horizontal axis, pi over 6. Now let's take a look at the graph of one of them. I want to talk about lines in polar coordinates.
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