My Favorite One-Liners: Part 9

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today, I’d like to discuss a common mistake students make in trigonometry… as well as the one-liner that I use to (hopefully) help students not make this mistake in the future.

Question. Find all solutions (rounded to the nearest tenth of a degree) of \sin x = 0.8.

Erroneous Solution. Plugging into a calculator, we find that x \approx 53.1^o.

arcsine1

The student correctly found the unique angle x between -90^o and 90^o so that \sin x = 0.8. That’s the definition of the arcsine function. However, there are plenty of other angles whose sine is equal to 0.7. This can happen in two ways.

First, if $\sin x > 0$, then the angle x could be in either the first quadrant or the second quadrant (thanks to the mnemonic All Scholars Take Calculus). So x could be (accurate to one decimal place) equal to either 53.1^o or else 180^o - 53.1^o = 126.9^o. Students can visualize this by drawing a picture, talking through each step of its construction (first black, then red, then brown, then green, then blue).arcsin45

However, most students don’t really believe that there’s a second angle that works until they see the results of a calculator.

TIarcsin45

Second, any angle that’s coterminal with either of these two angles also works. This can be drawn into the above picture and, as before, confirmed with a calculator.

So the complete answer (again, approximate to one decimal place) should be 53.1^{\circ} + 360n^o and 126.9 + 360n^{\circ}, where n is an integer. Since integers can be negative, there’s no need to write \pm in the solution.

Therefore, the student who simply answers 53.1^o has missed infinitely many solutions. The student has missed every nontrivial angle that’s coterminal with 53.1^o and also every angle in the second quadrant that also works.

green line

Here’s my one-liner — which never fails to get an embarrassed laugh — that hopefully helps students remember that merely using the arcsine function is not enough for solving problems such as this one.

You’ve forgotten infinitely many solutions. So how many points should I take off?

For further reading, here’s my series on inverse functions.

Merry Christmas!

Hosanna in Excel sheets.

hosanna-in-excel-sheetsAnd Merry Christmas.

 

Slide rule

To give my students a little appreciation for their elders, I’ll demonstrate for them how to use a slide rule. Though I have my own slide rule which I can pass around the classroom, demonstrating how to use a slide rule is a little cumbersome since they don’t have their own slide rules to use.

I recently found an applet to make this demonstration a whole lot easier: https://code.google.com/p/java-slide-rule/

Thoughts on Infinity (Part 3f)

In recent posts, we’ve seen the curious phenomenon that the commutative and associative laws do not apply to a conditionally convergent series or infinite product: while

\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ... = \ln 2,

a rearranged series can be something completely different:

\displaystyle 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} ... = \displaystyle \frac{3}{2} \ln 2.

This very counterintuitive result can be confirmed using commonly used technology — in particular, Microsoft Excel. In the spreadsheet below, I typed:

  • =IF(MOD(ROW(A1),3)=0,ROW(A1)*2/3,IF(MOD(ROW(A1),3)=1,4*(ROW(A1)-1)/3+1,4*(ROW(A1)-2)/3+3)) in cell A1
  • =POWER(-1,A1-1)/A1 in cell B1
  • =B1 in cell C1
  • I copied cell A1 into cell A2
  • =POWER(-1,A2-1)/A2 in cell B2
  • =C1+B2 in cell C2

The unusual command for cell A1 was necessary to get the correct rearrangement of the series.

Then I used the FILL DOWN command to fill in the remaining rows. Using these commands cell C9 shows the sum of all the entries in cells B1 through B9, so that

\displaystyle 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} \approx 0.961544012

log2series6

Filling down to additional rows demonstrates that the sum converges to \displaystyle \frac{3}{2}\ln 2 and not to \ln 2. Here’s the sum up to 10,000 terms… the entry in column E is the first few digits in the decimal expansion of \displaystyle \frac{3}{2} \ln 2. log2series7

Clearly the partial sums are not approaching \ln 2 \approx 0.693, and there’s good visual evidence to think that the answer is \displaystyle \frac{3}{2} \ln 2 instead. (Incidentally, the 10,000th partial sum is very close to the limiting value because 10,000 is one more than a multiple of 3.)

 

 

 

Thoughts on Infinity (Part 3d)

In recent posts, we’ve seen the curious phenomenon that the commutative and associative laws do not apply to a conditionally convergent series or infinite product. Here’s another classic example of this fact that’s attributed to Cauchy.

In yesterday’s post, I showed that

\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ... = \ln 2.

This can be (sort of) confirmed using commonly used technology — in particular, Microsoft Excel. In the spreadsheet below, I typed:

  • 1 in cell A1
  • =POWER(-1,A1-1)/A1 in cell B1
  • =B1 in cell C1
  • =A1+1 in cell A2
  • =POWER(-1,A2-1)/A2 in cell B2
  • =C1+B2 in cell C2
  • Then I used the FILL DOWN command to fill in the remaining rows. Using these commands cell C10 shows the sum of all the entries in cells B1 through B10, so that

1 - \displaystyle \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + \frac{1}{9} - \frac{1}{10} \approx 0.645634921

log2series1

 

Filling down to additional rows demonstrates that the sum converges to \ln 2, albeit very slowly (as is typical for conditionally convergent series). Here’s the sum up to 200 terms… the entry in column E is the first few digits in the decimal expansion of \ln 2.

log2series2

Here’s the result after 2000 terms:

log2series3

20,000 terms:

log2series4

And finally, 200,000 terms. (It takes a few minutes for Microsoft Excel to scroll this far.)

log2series5We see that, as expected, the partial sums are converging to \ln 2, as expected. Unfortunately, the convergence is extremely slow — we have to compute 10 times as many terms in order to get one extra digit in the final answer.

 

 

Deciphering recommendation engines

From the video’s description: “Data scientist Cathy O’Neil provides a glimpse of the methods that Netflix, Google, and others apply to recommend or offer to users selections based on their apparent interests.” This is a non-intuitive but real application of linear algebra.

Preparation for Industrial Careers in the Mathematical Sciences: Finding the Safest Place to Store Nuclear Waste

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the first pair of videos describing the process of mathematical modeling. From the YouTube descriptions:

Dr. Genetha Gray talks about her path and about a research problem that she worked on at Sandia National Laboratories. Using quite limited geological data, they had to create a groundwater flow computational model, with parameters to be determined, so that they could study the feasibility and safety of prospective subsurface nuclear waste storage sites.

Prof. Gwen Spencer of Smith College introduces the mathematics behind optimization, calibration, and the quantification of uncertainty in models and in the results that they give.

Arithmetic with big numbers: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on doing basic arithmetic with very large numbers that exceed the character displays of most calculators.

Part 1: Addition

Part 2: Multiplication

Part 3: Division

 

 

 

 

A clean computer programming joke

A programmer calls home. His wife says, “While you’re out, get some milk.” He never returns home.

Calculator Errors: When Close Isn’t Close Enough (Index)

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how I remind students about Taylor series. I often use this series in a class like Differential Equations, when Taylor series are needed but my class has simply forgotten about what a Taylor series is and why it’s important.

Part 1: Propagation of small numerical errors.

Part 2: A tragedy during the 1991 Gulf War that was a direct result of calculator rounding.