Tangent of angles approaching 90 degrees

Last week a colleague came to me with a puzzle. He asked me to punch in the tangent of 89 degrees into a nearby TI-83 calculator.

He asked me what was surprising about this number. I wasn’t surprised. I didn’t have an answer for him, although in retrospect I probably should have. He had to tell me this was the number of degrees in a radian. Oh! So it is.

Even further, he said, try punching in tan(89.9), or tan(89.99), etc.

Each one is (about) ten times the previous. (With the TI-83, replacing the last result with each new one, I didn’t see the “about” until later.) This is kinda neat! His question to me was: WHY is this true?

Tangent is a function that accepts an angle and spits out a ratio of lengths. It seems weird that the answer for tan(89) looks like a number of degrees. It is a unitless output, though, and ~57 degrees per radian is also unitless, so I suppose this isn’t much of an issue. The question is, why does it appear that

Ditching the degree measure,

My problem was in trying to first tackle this question visually.

tan(89 degrees) is the length of the vertical line lying between an extended radius of a unit circle drawn 89 degrees from the horizontal and the right side of the circle. The numbers above make it seem like it is 180/.

An equivalent image:

A triangle and circle times larger have the same relative lengths.

I tried to explain this by imagining rolling the circle over the tangent line, wrapping the line around the circle, etc. I didn’t get anywhere.

I also tried considering the fact that there’s nothing particularly special about degree measure, except for the fact that 360 is an easy to divide number. Does this happen with other angle units? For example, what about a unit that was, instead of 1/360 of a circle, a larger 1/100 of a circle? We could instead take the equation above and ask,

Is the tangent of one one-hundredth of a circle short of equal to the number of hundredths of a circle in a single radian? It looks to be true!

But this is only approximate. We could extend this to any fractional unit of a circle:

Using this different unit, where the approximation is less accurate, I was able to see that the degree version wasn’t exactly true, either. It definitely looks like dividing the circle into a larger number (360, rather than 100) yields a closer approximation:

https://www.desmos.com/calculator/vqbu4zni5p?embed

In the above, (red) and (blue) converge for larger (horizontal).

I was comfortable in concluding now that this wasn’t just a coincidence that relied on degree measure, and could extend this to include using 89.9, 89.99 etc degrees as well. In fact, tacking on .9s to the 1/nths of a circle units works. Just plugging in a bunch of numbers, it looks like

works for any n, and also extends to any power a, not just the integers.

https://www.desmos.com/calculator/ctcicykylj?embed

(red) and (green) lie almost on top of each other for positive (horizontal). In the link you can see this works for any by fiddling with a slider.

The question remained, why is this true? Now that I saw it’s only an approximation, I realized that I should be going about this algebraically from the start.

A trick, called the small angle approximation, is used in physics often to get rid of pesky sines and tangents when you’d rather just have an expression with the angle inside.

This behavior is clear when the functions are written in their Taylor series form:

When x is real small, all the higher power terms get super small, and the approximation becomes more accurate.

This approximation was my first thought, but there’s a problem: it works for small angles, but my colleague’s puzzle was about angles near 90 degrees. In fact, we can’t even fudge the Taylor series of tangent near here, because there is no Taylor series around 90 degrees. (This is a consequence of the fact that tan(x) blows up to infinity at 90 degrees.)

The problem is solved by noting that working with tangent near 90 degrees is the same as working with another trig function, cotangent, near 0 degrees.

Setting everything up:

Done! Having the instead of any old number is unneccessary — this works for any multiple. However, integer a‘s makes the trick of having the same digits show up in tan(89), tan(89.9), etc. work.

So, we can show this algebraically. I just wish I had a nice geometric argument.

More on the circular solution to the intersection of two perpendicular lines

In the last post, I showed that the intersection of two perpendicular lines must lie on a circle, so long as the lines are each forced to go through particular points. The final result was a parameterization based on the classic cosine, sine version of a circle, but the bit I found more interesting was the earlier form:

One of the results of the parameterization was that the angle at which the point lied on the circle was not the angle at which one of the lines made with the x axis (unless the circle’s center was at the origin). This led to a phase shift in the parameterization from the angle of the line. If we were willing to lose a bit of information, the phase, we could also show that the above is a circle if it satisfies

with

Since we already have the parameterization with m above, showing this is true is just a matter of algebra. To start, add and subtract the coordinates of the center of the circle from x and y:

Finding a common denominator and carefully combining gives

We now have a form that, when plugged into the LHS for the circle equation, cancels out the center point coordinates.

The RHS of this thing is a bit easier to work with with by letting

It becomes

Careful manipulation yields

Nicely, the m‘s cancel out completely. This becomes

Done!

Parameterizing a circle with the intersection point of two perpendicular lines.

I’ve been really taken with Desmos, an online calculator and easy to use graphing tool. My students have been using it for some time, and I’m especially happy with the “slider” tool that it offers. Whenever you put a letter into a function while graphing, it suggests a value to assign it, and lets you tune that value with the slider. This tool is similar to Mathematica’s Manipulate or Animate functions, which I’ve had success using in previous classes to show how a function depends on its parameters.

My year-long teacher’s Mathematica license recently expired, making it a bit tougher to install on a new device. While I do have access to an unsupported copy, Desmos has more than replaced M-ca for any of my presentation needs.

In a recent class, we were playing around with linear systems and intersecting lines. To show that a negative reciprocal slope leads to a perpendicular line, I assigned a slider to the value m, and made two linear equations with slopes of m and -1/m:

The slider has the nice effect of letting you rotate the lines to see that they’re always perpendicular.

Play with it yourself, why don’t ya. You can animate or adjust the slope with the m slider on the left.

The kids were delighted by the pinwheel spinning of the lines as the slope was adjusted. To show that we weren’t limited to lines that passed through the origin, I tacked on a y-intercept to both of the equations, and asked the students, what do you think happens when I adjust the slope now?

My point was to show that the lines remain perpendicular. I would have been pleased to hear that the students could also predict that the point of intersection of the two lines would now move around, instead of be fixed at the origin.

One student went further, however: he was able to predict that the point of intersection of the two lines will always be fixed to a circle.

You can adjust the slope once again, as well as the points the line are fixed to pass through using the sliders. Only adjusting the slope m keeps the point of intersection on a circle. You can also adjust the points the lines are forced through with the sliders below.

 

The student had seen a connection to his geometry class from the previous year. An inscribed angle is half the measure of the intercepted arc. An angle inscribing half the circle must then be a right angle. What this student had realized was the converse of this statement: that a right angle, formed by two perpendicular lines each forced to pass through particular points, must lie on a circle, and those two points are the endpoints of a diameter of that circle. I thought this was awfully insightful!

I figured it would be neat to try to show that this must be true on my own. Solving the system

gives the point

I thought this was really neat. We haven’t shown that this point lies on a circle yet, but assuming it does, it shows a way to parameterize a circle with m as the ratio of quadratics. Maybe this is something a mathematician would immediately recognize, but it’s new to me!

To show this does lie on a circle, I need to find an appropriate transformation that turns the above into the more familiar

for a circle of radius R and center . The obvious choice is to connect the slope of one of the lines to the angle on the circle:

The parameterization becomes

This is where all your trig identities pay off. Those denominators become squared secants, letting you get rid of the fractions altogether.

I’m having a bit of difficulty with formatting here. I’ll have to just write it like so:

The middle bits of these should pop out: a sine times a cosine is a part of one of the double angle formulas:

While we’re tossing in sine of a double angle, we might as well introduce the cosine of the double angle as well. This shows up from the squares:

Our parameterization becomes

What’s neat about this is that the center of the circle now falls out as a constant term at the end, and we’ve maintained some kind of symmetry with the sines and cosines.

Here’s where my trig knowledge stopped. The sine and cosines can be combined, though: a linear combination of sine and cosine should leave a single sine curve, but with a phase angle tossed in.

This is great! We’ve got a way to combine the a, b, c, ds to get something looking like a radius.

with

At this stage, we need to do is turn that first sine into a cosine (using .

We’re left with one remaining question: are the phase angles the same?

A couple more identities that I don’t have memorized clears this up:

This answers the question above: yes! Our parameterization is

If we wanted to make it a bit nicer, replace:

and we get a nice

a circle centered at x=(a+b)/2 and y=(c+d)/2. Woof! Bark bark! Woof woof bark!

Thoughts before watching Star Trek: Discovery

Looking back over my thoughts on Star Trek in general, I’m reminded that Star Trek: Discovery is the only TV show I’ve ever truly felt some investment in before its release. I do feel a sort of obligation to myself to pay attention to the show, to keep track of the details. It’s a very distinct feeling from the Abrams movies. In 2009, I had only watched TNG and TOS, and was also geared to expect a Star Trek movie, along the lines of the TNG movies, which taken all together are tolerable at best, and totally different in tone in the tv shows.

The new series has more incentive to live up to them if it is trying to cater to Star Trek fans, even if they can be nitpicky.

The Abbreviation

A lot of people are snickering about their abbreviation of the name of the show. Star Trek Discovery can be shortened to STD! Ha! Disease is funny! Didn’t the producers realize that, before deciding to use the forbidden fourth letter of the alphabet?

But really, this is a fine way to abbreviate the show’s name. I’m not going to tell you you can’t refer to the show as STD. We should remember, though, that there is an established abbreviation convention for the previous Star Trek series: we only abbreviate the subtitle! Star Trek (1966) is abbrevated TOS for The Original Series, Star Trek: The Next Generation has long been abbreviated TNG, Deep Space Nine DS9, Voyager VOY, Enterprise ENT. CBS’s use of DSC (or the more popular use of DIS) for Discovery is not a cover-up or a workaround, but falls in line with the rule.

The details and tech should serve the story

One of the big technical issues with Star Trek is in the speed of the ships themselves. Warp Factor, a number (1 through 9.995) that describes the ships faster-than-light speed, is a brilliant device because it draws the audience’s attention away from the actual speed of the ship, except for implying super-future-fast (warp 5) or flippin-fantastic-future-fast (warp 9). We don’t need to know how quickly the ships are zooming around in terms of kilometers per second for the sake of most stories. The show can be totally inconsistent from one episode to the next in terms of speeds and distances travelled and the times between, but very very few Star Treks are about the specific speeds of the ships. Stating these speeds has the potential to be distracting. It was great when the writers created a function and diagram mapping the warp factor to a real speed, but this would be violated many times afterwards.

The point to remember is that we are not going to get so caught up in this TV show that we’re going to lose sight of the fact that it isn’t real. We acknowledge the inconsistencies and move on, giving us mental energy to think about the story at hand.

This extends to other technologies or plot elements in the show that might seem contradictory, both to each other, or to our own world. Who cares that the Eugenics Wars of the 1990s, cited in the original series and The Wrath of Khan, didn’t seem to happen. Exhaustive in-universe retconning should only be done if it makes an interesting story, or else they’ll be wasting our time.

Some fans really griped that the ship bridge and technology featured in ST:DIS didn’t look like the original series. Again, the point of these visuals is to support a believable story, and not be distracting. For someone who has “lived in” the Star Trek universe their whole lives, thinking of Kirk on the bridge all the time, it might be distracting to try to believe that DIS comes ten years before and yet doesn’t have all the goofy memory tapes and buttons of the original series bridge. However, for most people, recreating the original Enterprise set for a modern television show that’s trying to take itself seriously would be much, much more distracting.

Regardless, it’s fun to nitpick and note details

A fan picking apart details and trying to fit everything together in the puzzle should be able to acknowledge they’re doing it for fun, or for their enjoyment of the show. There doesn’t need to be some higher purpose involved, and it will be in their best interest to learn how to do so without being angry or indignant towards the creators. It will also serve a consumer well to learn how to interpret someone else’s nitpicking and exploration as an exercise in puzzle-solving rather true criticism. I’m going to have fun recognizing Star Trek elements and seeing how they fit (or don’t!) in the established world.

“Missing” and tacked-on information

Fans have been skeptical about the use of Sarek in the show as Michael Burnham’s adopted Vulcan father. Sarek, introduced in the first series as Spock’s father, serves as the only character we’ve seen connecting Discovery to another Star Trek series. It’s an obscure (enough) character that only fans of Star Trek will really care about. And this is maybe a good point: using him shrinks the universe a bit. Why can’t we have a story about a bunch of people in the same universe who have never met an Original Series character? I’ve heard this argument made with the Aliens franchise, in which many incarnations of the movies, games, or comics seem to feature a family member of Ellen Ripley. It’s easy to be drawn from the narrative when presented with something unlikely. The reminder of a character from an earlier incarnation of Star Trek is designed to connect and bridge us more closely to the show, but it backfires by just reminding us that the show is manufactured to do so.

The fact that Spock has an adopted human half-sister, though, is not much of a retcon. Some fans have said, why did we never hear of Spock mention his sister Michael? Why didn’t Sarek, Perrin, or Sybok, Spock’s immediate family members, ever mention her in their appearances in their appearances in The Original Series, the movies, or The Next Generation?

I’d answer: because she never needed to come up. Spock is notorious for being efficient and logical. Do you think he’d yammer on about his experiences with his sister while in a life-or-death situation onboard the Enterprise? Even at the first appearance of Sarek, Spock appears reluctant to mention that he is his father, defaulting to the business at hand and referring to him as a diplomat. His failure to mention his sister is no plot hole. Although TNG season 7 might make it seem otherwise, a TV show does not need to have an episode devoted to each family member of each main character.

Again, let’s remind ourselves, this is a TV show. Let’s do our best to judge it as one, and not as an historical record of real events. We don’t have to convince ourselves to like it, completely disregarding continuity errors, but we can adjust how strongly we react.

 

 

Thoughts and speculation on Star Trek’s popularity

It may be possible that I have thought of Star Trek more than half of the days I’ve been alive. Those days are also pretty skewed towards the second half of my life as well. Is this much of a feat? Sometimes I will think of people or places that have been important in my life, and I’ll take pause to wonder how many weeks or months it’s been since I last thought of them. I’m not sure I have the same issue with this TV show. Plenty of entertainment is manufactured to stick in our minds.

People who like Star Trek can seem pretty evangelical for a bunch of humanists. This comes with any fandom, but it’s a special thing to be able to look at the culture surrounding the show that is often considered the originator of the modern fandom. It might be ridiculous to say that fanzines and fanfiction, slash fiction and shipping, conventions and cosplay were all created from this one 3-season show. People published and shared stories about their favorite characters beforehand, I’m sure. I’m also sure that plenty of these were a bit sexy, played with taboo, and featured author-insertion (no pun intended). Theatrically or historically minded folks dressed up and played parts for their own sake. Interestingly, one of the first major renaissance “faires” was in 1966, the year ST debuted, at the Paramount ranch (although Paramount did own the rights to broadcast the show until 1969). But Star Trek does seem to be a major popularizer for these ideas, and set up some expectations about things like anime conventions and unwritten rules about how to write yourself doing some cool kissing with Matt Smith without receiving too much judgement.

Part of the continuing success of Star Trek is due to its popularity. It’s a franchise, and success leads to success. It’s a household name.

There must be original things that originally led to its popularity, though, and keep it going. It boils down to the following.

It drew from established genres that were underserved and added to them. Gene Roddenberry drew inspiration for Captain Kirk from the Horatio Hornblower novels. I haven’t read these, but it seems that they themselves heavily draw from the fetishized “plucky” character type and high adventure of (boys’) Victorian-era novels and subsequent pulp fiction. In addition, the space-adventure theme of the show probably drew in a lot of grown-up boys who used to read Tom Swift, Flash Gordon, and still looked fondly back on watching Captain Video. The Twilight Zone, and other “short story” format science fiction shows existed beforehand, but it seems they didn’t have the draw of a character connection.

This is the awesome benefit of having a show based on an indefinitely fast space ship. You can have an episodic format which allows for missing a broadcast and still following the story, like a sitcom, but offers a way to feature completely different premises as often as the creators want. This gives more freedom than a sitcom, which returns to the status quo at the end of every episode, and also keeps the same neighbors around.

Star Trek wasn’t really a huge hit except with a vocal minority (you could spin as an example of the Pareto principle) until in syndication, after its cancellation. The Next Generation series wasn’t exactly a hit right off the bat either. But, funnily enough, it likely benefited from a similar drought of science fiction on tv, only a couple decades later. The drop in popularity (and as many others say, quality) of Star Trek in the 90s, as Deep Space Nine, Voyager and Enterprise continued on, was likely an indirect result of TNG’s popularity: the new Star Trek spawned a new interest in the genre. We might not have seen Babylon 5, Stargates, The X-Files without it, but these might have also saturated the market.

It straddled the lines of being interesting, silly, and thoughtful. Again, Star Trek wasn’t a huge hit at first, but the new-setting-every-week bit was attractive to the science fiction short story lovers. These types are usually looking for ideas that will stick with them for some time afterwards. The ending of The Twilight Zone I’m sure left a hole that Star Trek was able to fill. These ideas were social and political commentary, either unspoken or explicitly stated, or technological and scientific. Deeper personal stories wouldn’t start until the first movie, and The Next Generation, and the political intrigue wouldn’t really pick up until later seasons of TNG (and DS9!).

Viewers at least had some characters to count on week after week. Again, this could have attracted the pulp lovers who fondly remembered reading the Hardy Boys or Nancy Drew (although I’m not sure the Hardy Boys were giving the die-hard fans homoerotic suggestions in the first year of publications). These characters served as the much needed “human” connection that short stories tend to lack — having seen them week after week raises the stakes when they find themselves in peril.

The thinly veiled suggestions of a romantic or sexual relationship between Kirk and Spock brings us to the common assertion that Star Trek was intentionally camp. It’s come to my attention, after seeing Whatever Happened to Baby Jane? at our little Fancy Movie Night, that at the time the camp aesthetic was becoming more popular on television as well as in film. By the way, this might also owe itself to the popularity of Some Like it Hot, also seen at FMN, a film which served to help dismantle the Motion Picture Production Code. The effective removal of this explicitly written code made it a lot easier to play around with the potentially gay relationship, which allowed the creators to bring in other campy qualities.

The title of this section isn’t to suggest being gay is silly, but it may have appeared so to a typical viewer. In addition, the camp aesthetic definitely connected lightheartedness with homosexuality.

Finally, and maybe most obviously, the show was entertaining on a surface level. The traditionally boyish interests of seeing the ship fly around, shooting ray guns, goofy aliens, funny and sexy pajamas, and brawling action I’m sure solidified the interest of plenty of people (and not just boys).

But the thing people like to talk about the most nowadays is…

It introduced a progressive cast of skilled officers and regularly gave social commentary. This is the part that remains the most inspiring. I wonder how essential it was to the show’s initial popularity, but without these aspects, the show would certainly be a lot less interesting, and definitely not worth getting worked up over 50 years later. Having a collection of Earth’s races all working together on the same bridge, backed by Roddenberry’s vision that they have moved beyond interpersonal conflict, stands as a great source of inspiration. In addition to the obviously racial statements, the addition of the Russian Chekov in later seasons, although he was largely there for comic relief, represented a promising future (respect, as well as existence!) for what had been the United States’ chief enemy at the time. (I’ll have plenty of chance in the future to chat about the other representations of the USA’s enemies, the Klingons and the Romulans.)

Human progress was not only shown in social achievement or technological power. Spock himself was often played for laughs due to his rigidity and failure to acknowledge his (clearly present) humanity. Although loved and respected by fans, Spock often was a straw man representing the emotionally bankrupt, completely unempathetic man struggling to maintain feelings of superiority based on knowledge and critical thinking. I think if Spock were a character created today, he would be compared to “mansplainers” or Red Pill types. The fact that the other characters can explain their choices outside of Spock’s coldness, and make fun of him when he lets a bit of emotion shine through, promises a future in which people don’t have to be miserable pedants who only care about things for their academic value. The human still has a place in the 23rd century. We don’t have to replace ourselves or mimic machines as time pushes forward. People liked this idea.

 

I’ve been thinking about Star Trek even more than usual lately, what with the first episodes of Star Trek Discovery airing last week. I’ve got even more to say about that, but perhaps it can wait for another day.

A Note to Myself

Baader-Meinhoff ≠ Bernie Madoff ≠ Walter Matthau.

The Image Superiority Effect and The Baader-Meinhof Phenomenon

Last year I came across what was at the time a popularly shared image listing cognitive biases (found at the bottom of this Medium article). The graphic originally reminded me of An Illustrated Book of Bad Arguments, which I had picked up along with Randall Munroe’s Thing Explainer, maintaining a general theme with my purchase. The Book of Bad Arguments unfortunately didn’t turn out so great: the illustrations are sort of cute sometimes, but are more often a bit awkward and rarely illuminated the logical fallacy they were attempting to explain. In addition, the written explanations themselves were not brilliant, and I’ve also come to find that a couple descriptions were just plain wrong.

I do highly recommend the Thing Explainer. It takes the cute and informative style of Munroe’s Up Goer Five and fills a book with it, and with more detail as well.

Anyway, I saved the cognitive bias diagram on my computer desktop, and it has been sitting there since. A few days ago I opened the image and scanned around it, and saw the Image Superiority Bias. The basic idea is that we tend to remember information given to us as images more easily than information given in words. From an “amount of information” perspective, this is a bit bewildering. Compare a jpeg file to a text document: one might say an image is worth a thousand words, but the first is going to be a few MB while the other a few kB. The idiom reflects the actual way our brains work. From a biological perspective, maybe this makes sense. Interpretation of vision is an ancient process for animals, while interpretation of text or spoken word is an extremely new concept, and we still have to spend a decade or two practicing to get any good at it.

Our ability to understand and classify images quickly makes it feel like there’s a paradox in computing: my PC can easily store the text of tens of thousands of books, but until very recently computers have been pretty unreliable at pattern recognition. This is why projects like Zooniverse exist, in which users are asked to answer pretty simple questions about objects in a vast database of photos.

Computers aren’t so great at classifying galaxies, and have a tough time differentiating spiral galaxies from elliptical. Humans are real good at it. I can even tell you that the spiral arms come out clockwise from the center! (galaxyzoo.org)

Later in the day I was trying to think of a certain website that had lots of user-submitted Do It Yourself project descriptions. I couldn’t think of the name, but I DID remember that it had a funny hand-drawn robot as a mascot.

Which is easier to remember?

“Instructables” is not a difficult name to remember. But that yellow robot sure stuck in my head more easily. I thought back on the image superiority bias I had just read about that morning.

This experience reminded me of another bias, The Baader-Meinhof phenomenon, in which it can seem like something you just learned about is suddenly popping up everywhere in your life. Sometimes the thing is actually a new song or cultural idea or whatever, and so the time you learn about it and the instances it starts showing up in your life would naturally coincide. However, this effect also happens with not-necessarily-new, but newly-learned things as well. For instance, you may have heard references to an historic figure over and over since learning about them a few weeks ago. In fact, these references might have been coming up anyway, but we have a way of filtering out a sizable chunk of the information we’re given all the time, especially that which we aren’t familiar with.

The fact that I had read about the image superiority bias just that morning, and noticed an instance of it that day, felt like a mix of observer bias and the Baader-Meinhof effect.

I tend to associate the B-M effect with the idea of Maslow’s Hammer, the idea that it’s easy “if the only tool you have is a hammer, to treat everything as if it were a nail.” In both cases,  we preferentially deal with things in the ways I have dealt with them in the past, and ignore those ways and things we’re not familiar with. Unless we’re willing to constantly experiment with everything (inevitably ruining everything in the process), we have to live this way to some degree.