My Favorite One-Liners: Part 106

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.

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}},

\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}},

\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}.

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

My Favorite One-Liners: Part 104

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.

I use today’s quip when discussing the Taylor series expansions for sine and/or cosine:

\sin x = x - \displaystyle \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots

\cos x = 1 - \displaystyle \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots

To try to convince students that these intimidating formulas are indeed correct, I’ll ask them to pull out their calculators and compute the first three terms of the above expansion for $x=0.2$, and then compute \sin 0.2. The results:

This generates a pretty predictable reaction, “Whoa; it actually works!” Of course, this shouldn’t be a surprise; calculators actually use the Taylor series expansion (and a few trig identity tricks) when calculating sines and cosines. So, I’ll tell my class,

It’s not like your calculator draws a right triangle, takes out a ruler to measure the lengths of the opposite side and the hypotenuse, and divides to find the sine of an angle.


My Favorite One-Liners: Part 63

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.

I’ll use today’s one-liner to explain why mathematicians settled on a particular convention that could have been chosen differently. For example, let’s consider the definition of  y = \sin^{-1} x by first looking at the graph of f(x) = \sin x.


Of course, we can’t find an inverse for this function; colloquially, the graph of f fails the horizontal line test. More precisely, there exist two numbers x_1 and x_2 so that x_1 \ne x_2 but f(x_1) = f(x_2). Indeed, there are infinitely many such pairs.

So how will we find the inverse of f? Well, we can’t. But we can do something almost as good: we can define a new function g that’s going look an awful lot like f. We will restrict the domain of this new function g so that g satisfies the horizontal line test.

For the sine function, there are plenty of good options from which to choose. Indeed, here are four legitimate options just using the two periods of the sine function shown above. The fourth option is unorthodox, but it nevertheless satisfies the horizontal line test (as long as we’re careful with \pm 2\pi.

sine2So which of these options should we choose? Historically, mathematicians have settled for the interval [-\pi/2, \pi/2].

So, I’ll ask my students, why have mathematicians chosen this interval? That I can answer with one word: tradition.

For further reading, see my series on inverse functions.



My Favorite One-Liners: Part 40

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.

In some classes, the Greek letter \phi or \Phi naturally appears. Sometimes, it’s an angle in a triangle or a displacement when graphing a sinusoidal function. Other times, it represents the cumulative distribution function of a standard normal distribution.

Which begs the question, how should a student pronounce this symbol?

I tell my students that this is the Greek letter “phi,” pronounced “fee”. However, other mathematicians may pronounce it as “fie,” rhyming with “high”. Continuing,

Other mathematicians pronounce it as “foe.” Others, as “fum.”

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.


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.


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.

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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.

My Favorite One-Liners: Part 8

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.

At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

  1. Arithmetic/Algebra: a \cdot (b + c) = a \cdot b + a \cdot c. This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
  2. Algebra: If a,b > 0, then \sqrt{ab} = \sqrt{a} \sqrt{b}.
  3. Algebra: If a,b > 0 and x is any real number, then (ab)^x = a^x b^x.
  4. Precalculus: \displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i.
  5. Precalculus: \displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i.
  6. Calculus: If f is continuous at an interior point c, then \displaystyle \lim_{x \to c} f(x) = f(c).
  7. Calculus: If f and g are differentiable, then (f+g)' = f' + g'.
  8. Calculus: If f is differentiable and c is a constant, then (cf)' = cf'.
  9. Calculus: If f and g are integrable, then \int (f+g) = \int f + \int g.
  10. Calculus: If f is integrable and c is a constant, then \int cf = c \int f.
  11. Calculus: If f: \mathbb{R}^2 \to \mathbb{R} is integrable, \iint f(x,y) dx dy = \iint f(x,y) dy dx.
  12. Calculus: For most differentiable function f: \mathbb{R}^2 \to \mathbb{R} that arise in practice, \displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}.
  13. Probability: If X and Y are random variables, then E(X+Y) = E(X) + E(Y).
  14. Probability: If X is a random variable and c is a constant, then E(cX) = c E(X).
  15. Probability: If X and Y are independent random variables, then E(XY) = E(X) E(Y).
  16. Probability: If X and Y are independent random variables, then \hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y).
  17. Set theory: If A, B, and C are sets, then A \cup (B \cap C) = (A \cup B) \cap (A \cup C).
  18. Set theory: If A, B, and C are sets, then A \cap (B \cup C) = (A \cap B) \cup (A \cap C).

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

  1. Algebra: (a+b)^x \ne a^x + b^x if x \ne 1. Important special cases are x = 2, x = 1/2, and x = -1.
  2. Algebra/Precalculus: \log_b(x+y) = \log_b x + \log_b y. I call this the third classic blunder.
  3. Precalculus: (f \circ g)(x) \ne (g \circ f)(x).
  4. Precalculus: \sin(x+y) \ne \sin x + \sin y, \cos(x+y) \ne \cos x + \cos y, etc.
  5. Precalculus: \displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right).
  6. Calculus: (fg)' \ne f' \cdot g'.
  7. Calculus \left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}
  8. Calculus: \int fg \ne \left( \int f \right) \left( \int g \right).
  9. Probability: If X and Y are dependent random variables, then E(XY) \ne E(X) E(Y).
  10. Probability: If X and Y are dependent random variables, then \hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y).

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

green lineI wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”


What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 18

The Riemann Hypothesis (see here, here, and here) is perhaps the most famous (and also most important) unsolved problems in mathematics. Gamma (page 207) provides a way of writing down this conjecture in a form that only uses notation that is commonly taught in high school:

If \displaystyle \sum_{r=1}^\infty \frac{(-1)^r}{r^a} \cos(b \ln r) = 0 and \displaystyle \sum_{r=1}^\infty \frac{(-1)^r}{r^a} \sin(b \ln r) = 0 for some pair of real numbers a and b, then a = \frac{1}{2}.

As noted in the book, “It seems extraordinary that the most famous unsolved problem in the whole of mathematics can be phrased so that it involves the simplest of mathematical ideas: summation, trigonometry, logarithms, and [square roots].”

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When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

Sine of madness

sine of madness

A natural function with discontinuities (Part 3)

This post concludes this series about a curious function:

discontinuousIn the previous post, I derived three of the four parts of this function. Today, I’ll consider the last part (90^\circ \le \theta \le 180^\circ).

obtuseangleThe circle that encloses the grey region must have the points (R,0) and (R\cos \theta, R \sin \theta) on its circumference; the distance between these points will be 2r, where r is the radius of the enclosing circle. Unlike the case of \theta < 90^\circ, we no longer have to worry about the origin, which will be safely inside the enclosing circle.

Furthermore, this line segment will be perpendicular to the angle bisector (the dashed line above), and the center of the enclosing circle must be on the angle bisector. Using trigonometry,

\sin \displaystyle \frac{\theta}{2} = \frac{r}{R},


r = R \sin \displaystyle \frac{\theta}{2}.

We see from this derivation the unfortunate typo in the above Monthly article.

Engaging students: Graphing Sine and Cosine Functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Emma Sivado. Her topic, from Precalculus: graphing sine and cosine functions.

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D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

First, I would pose the question “how did the Egyptians build the pyramids without calculators without measuring tapes and without the advanced mathematics we have today?” After a short discussion I would ask them if we want to build a pyramid that is 250 meters high and the base is 360 meters long how long would we need to make the hypotenuse? Already knowing the Pythagorean Theorem the students would be able to answer the question. Then, I would tell them that historians have found Egyptian scribes asking questions such as these in order to build the pyramids, and systems of ropes with knots were used to measure lengths. These relationships in right triangles created the sine and cosine functions we know today. Sine and cosine date back to 1900 BC where they were used to calculate angles in order to track the motion of the planets and stars. However, the definition of sine and cosine in terms of right triangles was not recorded until 1596 AD by Copernicus.


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A.2: How could you as a teacher create an activity or project that involves your topic?

I found a great activity that encompasses all of the aspects of graphing sine and cosine on the University of Arizona website. Depending on how transformations in the linear and quadratic functions were introduced, this activity could follow the same pattern; allowing the students to explore the ideas themselves and having them put the content into their own words. The activity begins by giving an example of a bug walking on an upright loop. The instructor asks the students what the graph would look like of the bug’s distance from the ground vs. time. I would probably use a different, more concrete example because there are plenty of things the students know that go around in circles. The best example I think is a Ferris wheel. So after the students are able to tell you what the graph would look like you relate that to the unit circle and how the sine and cosine functions follow the same pattern of going around the circle counterclockwise. Next, you let the students plot points from the unit circle onto the Cartesian plane showing them that their prediction was correct; the sine and cosine functions make a wave. Now that they have drawn the parent function you let them explore the functions f(x) = asinx or f(x)= acosx, then f(x) = sin(bx) or f(x) = cos(bx), then finally f(x) = sin(x+c) or f(x) = cos(x+c) to let them discover how a, b, and c change the amplitude, period, frequency, and starting point of the graphs.

This is a great activity because the students use multiple examples to see how a, b, and c affect the parent graph of sine and cosine. The activity promotes inquiry based learning and will help deepen the understanding of the graphs of sine and cosine.



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E.1: How can technology (YouTube, Khan Academy [], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Math can be seen in many forms of art from music to painting. I remember one of my favorite activities from math in high school was creating pictures with sine and cosine functions. We were able to draw flowers, clovers, and hearts simply with only the sine and cosine functions. After the students understand the parent function you can give them an exploration activity on their graphing calculator where they plug in various sine and cosine functions to draw flowers, clovers, and hearts. After that challenge the students to draw their own picture using the patterns they see from the examples. These same ideas can be used in computer graphics and animation to draw similar figures, and a lot of students are interested in computers and especially video games so this should be a fun activity for them.