On Trig and Cosby

cos b

The man on the right is Bill Cosby, an actor and comedian who became a staple of Thursday night television for NBC from the mid-1980’s to the early 1990’s. (More recently, he has become embroiled in controversy because of allegations regarding extremely unsavory actions in his past.  That’s not relevant to this joke, but it probably limits the joke’s “re-tell” value.) His last name is pronounced “KAHZ-bee”, which is important to know if you want to get the joke.

The rest of this image is all about trigonometry. You remember trigonometry, right? Maybe? No? Okay, let’s have a refresher.

Imagine a right triangle. A right triangle has one ninety-degree angle, and two angles that are both less than 90º. Let’s call these angles a, b, and c, with c being the right angle.


Now for the sake of convenience, I’m going to label the sides A, B, and C. Side A will be directly opposite angle a, and so on.


Side C, which is opposite angle c, is the hypotenuse. The hypotenuse of a right triangle is always opposite the 90º angle, and it is always the longest of the three sides.

Sides A and B are known as legs. There is a web of relationships between the measures of angles a, b, and c, and the lengths of sides A, B, and C.

Consider angle b, in the lower left corner of the triangle.


From the perspective of angle b, side B is opposite and side A is adjacent, meaning that side A is one of the legs that forms angle b. Side C is the hypotenuse, as always, and also encloses angle b. Clear as mud? Good.


To fully understand this joke, you need to know about three basic trigonometric functions: sine, cosine, and tangent. These functions are often abbreviated sin, cos, and tan. The sine of an angle is found by dividing the length of the opposite side by the length of the hypotenuse. For angle b, that’s side B divided by side C.

A sine of things to come.

A sine of things to come.

The cosine of angle b is found by dividing the length of the adjacent side (side A) by the length of the hypotenuse (side C).


The tangent of angle b is found by dividing the length of the opposite side (side B) by the length of the adjacent side (side A). It does not involve the hypotenuse at all.


There’s another interesting relationship between sine, cosine, and tangent. The tangent of angle b is equal to sin b divided by cos b. I’ll leave it as an exercise for the reader to prove that.


Now we’re in the home stretch. Multiplying both sides by cos b gives:


And dividing both sides by tan b gives:

cos b



Bechdel Jest

Two women walked into a bar and discussed the Bechdel test.

You know, it’s hard to think of another one-liner that is so charmingly self-referential.

The criteria that would eventually be known as the Bechdel test were first described in 1985 in a comic strip called Dykes to Watch Out For, by Alison Bechdel. In one particular strip (“The Rule“) two women are discussing the possibility of seeing a movie. One of the women says that she avoids any movie that fails to satisfy three criteria:

  1. The movie must have at least two female characters, who
  2. Talk to each other, about
  3. Something other than a man.

She then quips that she has been unable to watch any movie since Alien, since the two female characters in Alien talk to each other about the monster.

These criteria have since been adopted for evaluating the gender bias in any sort of entertainment media, including video games, television shows, books, etc. How many films pass the test? According to bechdeltest.com, a user-edited database of more than five thousand movies, roughly 56% of the movies listed pass all three of the test’s criteria (it should be noted that bechdeltest.com requires that both female characters be named).

And now we can add at least one joke that passes the Bechdel test with flying colors. Bravo, anonymous joke author. Bravo.

Solipsism Silliness

Is it solipsistic in here, or is it just me?

Here’s a bit of philosophical nerdiness for you. Solipsism is the idea that nothing can be known to exist but the individual’s consciousness. When I say “the individual”, I mean you…or me…or whoever. According to a solipsist, everything outside your head must be filtered through your consciousness. In fact, the external world might even be a product of your consciousness. We can only be certain of our own consciousness because we experience it directly. Rene Descartes said it best: I think, therefore I am. I can satisfactorily determine the existence of my own mind, but everything else could be an illusion.

A solipsistic viewpoint is untestable (which is why it falls under the purview of philosophy rather than science). Any test you could devise to determine whether the external world was real would necessarily be filtered through your consciousness. If your own mind was insistent on maintaining the illusion of an external universe, then your test results would be utterly useless, as your mind would mold them into your perceived reality.

While a person may be philosophically solipsistic, most of us behave as empiricists; that is, we act as if we believe in the objective reality of an external world, complete with physical rewards to be won, dangers to be avoided, and consequences for our actions. We do this because it yields consistent results. In most cases, our interactions with the physical world proceed as expected. Many would argue that if the external world (real or illusory) is indistinguishable from an empiricist philosophy, then there’s little point in debating whether it’s real or not. Maybe not, but it’s still kind of interesting (and maybe a bit unsettling) to imagine that reality begins and ends in your own mind, and that the universe as we know it…excuse me, as I know it, may be nothing but a dream.

Programming Problems

The programmer’s wife says: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”

The programmer comes home with 12 loaves of bread.

I’ve got to be honest here: I have very limited programming experience and probably cannot dissect this joke as thoroughly as a more seasoned programmer would. Nevertheless, I think it’s a cute joke. It’s definitely worth a mention and a cursory explanation.

I do know enough about programming to know that the order and phrasing of instructions is of utmost importance. The programmer’s wife opened with an instruction to get a loaf of bread, then followed with a conditional test: if they have eggs. A conditional test can be either true or false. If the test evaluates true, then the set of instructions immediately following the test is to be carried out.

Unfortunately, the wife left her instructions ambiguous; she told her husband to get a dozen, but didn’t specify what he was to get a dozen of. A normal human would have inferred that he should get a dozen eggs, not a dozen loaves of bread, but a computer is incapable of inferring meaning from ambiguous instructions. Interpreting the instructions in the same way a computer might, the husband assumed that the term dozen referred back to the loaf of bread. Now they can have sandwiches for months, but no omelets.

Cool Story BrO-

Q. Want to hear a joke about sodium?

A. Na.

Q. Want to hear a joke about sodium hypobromite?

A. NaBrO.

Chemists deal with lots of strange-sounding materials like sodium, ytterbium, and potassium hexacyanoferrate(III), so they’ve developed a system of symbols and formulas to represent these materials. The elements of the periodic table all have one- or two- letter symbols*. In some cases the symbol obviously descends from the element’s name; to wit, the symbol for oxygen is O, the symbol for titanium is Ti, and the symbol for chlorine is Cl.

Some elements – particularly the elements that have been known since antiquity – have symbols that do not seem logically connected to their name. In these cases, the symbol usually refers to an older name for the element, perhaps from Latin or Greek. For example, the symbol for sodium, Na, comes from the Latin word natrium. The word natrium has an even deeper, richer history among the alchemists and scholars of the ancient world, or so says Elementymology & Elements Multidict, a most fascinating site if you’ve got a few minutes to spare. At any rate, in 1807, Sir Humphrey Davy isolated the metal and recognized it as a distinct element. Bucking centuries of tradition, Sir Davy decided the metal ought to be named sodium, for he isolated it from caustic soda (now known as lye or sodium hydroxide).

So what about sodium hypobromite? Sodium hypobromite is a compound that contains sodium, bromine, and oxygen. Bromine and oxygen combine to form a group called a polyatomic ion. Since this group has an overall negative charge, it bonds quite easily with positively-charged sodium ions.

Bromine is fairly versatile as far as bonding with oxygen goes; there are actually several polyatomic ions made of bromine and oxygen. They are distinguished by the number of oxygen atoms bonded to the bromine atom, and by small variations in their names. Here are four compounds that can be made from the same three elements, and their names:

  1. NaBrO = sodium hypobromite
  2. NaBrO2 = sodium bromite
  3. NaBrO3 = sodium bromate
  4. NaBrO4 = sodium perbromate

If one isn’t too picky about capitalization, one could read NaBrO as “Na, bro”, meaning “No, close male associate (or perhaps my biological brother), I do not wish to hear a joke concerning sodium hypobromite.” Of course, the person already indicated his disinterest in hearing a joke about sodium. Perhaps this fellow simply isn’t in the mood for chemistry-themed jokes. We’ve all had days like that.

*Some periodic tables have three-letter symbols for the last few elements. These symbols represent temporary names. When the International Union of Pure and Applied Chemistry, or IUPAC, assigns permanent names to these elements, they will get a traditional one- or two-letter symbol to go along with their name.

Chemists and Plumbers

Q. How can you tell the difference between a chemist and a plumber?

A. Ask them to pronounce “unionized”.

Yeah, that might work. Here are some other nifty ways to tell the difference between a chemist and a plumber.

  1. Ask “Are you a chemist or a plumber?”
  2. Observe their appearance: plumbers wear coveralls and carry large pipe wrenches; chemists wear lab coats and carry beakers.
  3. Where did you see them? Plumbers come to your home in trucks, then shimmy through the crawl space. Chemists are usually found in laboratories. They seldom visit your home on business and they almost never shimmy.

When asked to pronounce the word unionized, a plumber would probably say “YOON-yun-ized”, as in belonging to a labor union. Labor unions are worker advocacy groups and are present in many professions. They try to protect worker rights and settle disputes between the labor and management.

A chemist would be more likely to say “un-EYE-on-ized”. In order to understand the meaning of unionized, you have to know the meaning of ionized. A molecule or atom is ionized when it acquires a positive or negative charge. This can happen when it loses or gains electrons. We’ve talked about this before. Acids and bases can also become ionized, not by gaining or losing electrons, but by splitting into charged pieces.

Consider hydrogen chloride, HCl. In its pure form, HCl is a noxious, colorless gas. The hydrogen atom and the chlorine atom share a pair of electrons; a covalent bond. The molecule as a whole is neutral, meaning it has an equal number of protons and electrons and no net charge.

When HCl dissolves in water, it changes. The hydrogen separates from the chlorine, but chlorine keeps both of the shared electrons. This leaves the hydrogen ion bereft of electrons, resulting in an overall positive charge. The chloride ion, for its participation, acquires a net negative charge. In this way HCl splits into ions without actually gaining or losing electrons.

HCl → H+ + Cl

The liberated H+ ions are absorbed by water molecules, resulting in the formation of another ion: the hydronium ion, H3O+. Hydronium ions are the defining characteristic of aqueous acidic solutions. Any acid, dissolved in water, will yield hydronium ions.

H+ + H2O → H3O+

Bases also react with water molecules to produce ions. Instead of hydronium ions, bases produce hydroxide ions, OH. Consider the reaction of ammonia, NH3, a base, with water:

NH3 + H2O → NH4+ + OH

But we’re not talking about ionized molecules, we’re talking about unionized molecules. That part’s simple: an unionized molecule is any molecule that hasn’t been ionized. Yeah, I know: I could have just said that from the start. But you’ve got to have the foundation before you can get the joke. Right?

A Pascal Pun

Albert Einstein, Isaac Newton, and Blaise Pascal are playing Hide ‘n’ Seek. It’s Einstein’s turn to count, so he covers his eyes and counts to ten. Pascal runs to hide, but Newton draws a one meter by one meter square on the ground, then stands in the middle of it.

Einstein reaches ten and uncovers his eyes. He sees Newton immediately and exclaims “I found you, Newton! You’re it!”

Newton replies “You didn’t find me. You found a Newton over a square meter. You found Pascal!”

Oh, that rascally Newton. Or should I say, that Pascally Newton?

No, I probably shouldn’t.

Anyway, let’s talk about pressure. Not the emotional pressure of having to meet a deadline, or peer pressure, but fluid pressure. Fluids, like air or water, are drawn towards the center of Earth just like everything else on this planet due to the influence of gravity. These fluids exert their weight on anything beneath them. We experience this weight as pressure.

Unlike the weight of a solid object, pressure doesn’t just push downward; it pushes in all directions. Here, at the bottom of Earth’s atmosphere, you have almost fifteen pounds of force pushing inward on every square inch of your body. A square inch is roughly the area of a postage stamp, so the average human body has a lot of square inches to it. Using the Du Bois formula, we can estimate that a 75-kilogram man (165 pounds) who is 178 cm tall (5′ 10″) ought to have a body surface area of about 1.9 square meters (about 3000 square inches). If that man is at or near sea level, he’ll have nearly 20 metric tons (22 short tons) of force pushing inward.

Now hold on, I hear you saying. If we’re all subject to multiple tons of force, pushing inward from all directions, why don’t we all get squished like bugs?

There are several reasons:

  1. We’re adapted to survive under this pressure.
  2. We’re full of fluids that are pushing outward with equal force.
  3. Most of the stuff inside us is fairly incompressible anyway.

So don’t worry too much about it. Just think about how amazing you are for standing up to that kind of pressure. Go you.

What do Newton, Einstein, and Pascal have to do with any of this? Well, for all of his accomplishments, Einstein is not really necessary to this joke. You can replace him with your favorite scientist; say, Alfred Wegener.

A newton (after Sir Isaac Newton, in case there was any doubt) is the metric unit of force (not pressure, and it’s important to make this distinction!) A force is a push or pull, and for all intents and purposes a force acts on a single point. Pressure, on the other hand, is a force spread out over an area. When you walk about pressure using metric units, you talk about newtons per square meter.

The metric unit of pressure is the newton per square meter, or N/m2. There’s another, shorter name for this unit: the pascal (abbreviated Pa). One pascal is exactly equal to one newton per square meter, and the two terms are used interchangeably. So, by standing on an area of one meter by one meter (a square meter), Newton made himself a Pascal. Cute.