Problems: Trigonometry

ProblemArcCosThe website TeXample.net has a diagram which I modified to produce the image above. The problem is to find the length of the belt in terms of $latex \cot(\theta)$. I've added this to the Problems page.

You'll need to recall a little geometry: the belts between the two pulleys are tangent to each circle. In addition, there's some symmetry which gives us a total of 4 angles with degree $latex \theta$.

Odds and Ends: October 23, 2013

UnitCircleIn looking for material to use in trigonometry I found some code posted here on the Sage math website. The code, for a Sage interact manipulative by Jurgis Pralgauskis, seemed better suited for an animated GIF. I added a little extra code and put Sage to work. I waited--and waited--and waited; on the order of 40+ minutes with a quad core computer. The result is a nice animated GIF you can find on the Sage Output page. A screenshot is posted above. A radians version would be more useful to me; perhaps when I have more time.

Three links have been added to the sidebar:

  • Sage math Facebook page
  • Sage math Twitter feed
  • Web Equation

You might remember the Web Equation link broke some time ago. It's a useful tool, so I'm happy to be able to put it back on the list of links.

How do I...get the picture where I want?

FloatIt's a complaint I've heard quite a few times: why doesn't LaTeX put the figure where I've told it? Now most of the time $\LaTeX$ does a good job but sometimes it ignores your pleas to put the picture where you've instructed; it's found a better place, even if that place is unacceptable to you. There are various ways to override the behavior; I've suggested the float package. It's simple and has worked very well for me. You can find the new page listed on the sidebar or you can just click here.

Why do we need radians?

When you teach unit circle trigonometry you've got to introduce radians, and inevitably that leads to problems. Students aren't initially comfortable with radians and you'll almost certainly have someone ask why they need to use radians.

To be sure, the degrees/minutes/seconds system only described a finite number of angles but with decimal degrees, there is the required infinite number of angles needed to map each degree to a point on the unit circle. For any angle in radians we can simply convert to degrees using the conversion factor $latex \frac{180^{\circ}}{\pi}$; why then do we need radians?

You can address this natural question by including a simple math model into your lesson. The displacement of a weight attached to a spring moves up and down over time (non damping) according to some equation like $latex y=\frac{1}{5}\cos(4t)$. Physclips is a good site for animations; check out this page to see spring movement juxtaposed with movement along the sine curve. You can download clips to your computer, the harmonic motion example is found here. After seeing how the equation models reality you're ready for the teaching point: Radians are necessary because mathematical models use trig functions where the variable is time, which is not measured in degrees.