A Postcard from Spokane

A brief aside from the current series on general relativity — and the mysterious 43 seconds of arc per century in Mercury’s orbit — that turned into further discussion about angle measurement.

A few months ago, I received this clever postcard from someone visiting Spokane, Washington. The sender clearly knew the recipient (me) well: rather than sending me a postcard showing the jaw-dropping beauty of the Spokane area, I was impressed with the mathematical precision given for Spokane’s location.

I started wondering about exactly how precisely the postcard was measuring the location of Spokane — was it the location of City Hall or some other important landmark? — and I went to Google Maps to find out. (For what it’s worth, xkcd had a comic about this some time ago.)

And then it finally hit me, after far longer than it should have taken, that the postcard is utterly nonsensical.

We would never say that someone’s height is 4 feet, 20 inches. There are 12 inches in a foot, and so we would instead say that the height is 5 feet, 8 inches.

Likewise, when specifying an angle with minutes and seconds, there are (just like with ordinary time) 60 seconds in a minute and 60 minutes in a degree (so that there are 3600 seconds in a degree). Therefore, specifying an angle with 67′ or 66″, as in the postcard, makes absolutely no sense.

Furthermore, if converted into standard notation, we obtain a location of $48^{\circ} 7' 36''$ north, $117^{\circ} 42' 6''$ west, which is about 40 miles NNW of Spokane. (Images made by https://www.gps-coordinates.net/). Note on the conversion into decimal:

$47 + \displaystyle \frac{67}{60} + \displaystyle \frac{36}{3600} = 48 + \displaystyle \frac{7}{60} + \displaystyle \frac{36}{3600} = 48.12666\dots$

and

$117 + \displaystyle \frac{41}{60} + \displaystyle \frac{66}{3600} = 117 + \displaystyle \frac{42}{60} + \displaystyle \frac{6}{3600} = 117.701666\dots$

It’s a shame that the designer of the postcard made this error, as I genuinely thought this was a clever and aesthetically pleasing design idea for a postcard.

While I’m not sure how this mistake happened, my best guess is that the designer used the location of $47.6736^\circ$ north, $117.4166^\circ$ west — which is indeed in Spokane — and then misconverted from decimal notation to minutes and seconds.

What is your go-to dance move?

This is frivolous but fun, and it made me smile. Well done.

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Snell’s Law and a mystery novel

Lately, for my own leisure reading, I’ve been enjoying the murder-mystery novels of Dorothy Sayers. Her books are an enjoyable trip back in time, as she paints a very vivid portrait of English life of during the interwar years of the 1920s and 1930s. (Of course, at the time she was writing, no one had any idea that the Great War would not actually be the war to end all wars, as was the popular sentiment of the time.) Indeed, her first novel was published literally a century ago in 1923. The lead character, Lord Peter Wimsey (back then, the aristocracy was still part of English culture), has a distinctive way of speaking that makes the novels so delightful. A hallmark of the Sayers novels is that she didn’t merely write whodunit stories; instead, she strove to write novels in which a detective story happens to happen.

As an aside, I learned in her novel Gaudy Night that the adjective Oxonian means “related to Oxford,” which led me to further learn that my hometown of Oxon Hill, Maryland was so named because somebody, centuries ago, thought that the landscape of that part of the state reminded him of Oxford, England. While that comparison might have been reasonable centuries ago, it certainly would raise eyebrows today.

Anyway, with all that as background, in her story Unnatural Death, the following figure depicts an aerial view of a witness’s testimony at a key point in the story. I think I can describe this much of the scene without giving away the plot: the witnesses stood just inside the door of elderly Miss Dawson’s bedroom. A screen blocked direct observation of Miss Dawson as she lay in bed, but the witnesses could see Miss Dawson in the mirror.

As I read the novel, I immediately noticed that the mirror in the figure was not a perfect reflector… at the mirror, the angles of reflection of the dashed path of light are quite different. Indeed, I pulled out my protractor: the angle where the word “Mirror” is located has a measure of about 52 degrees, while the opposite reflected angle has a measure of about 72 degrees.

As this is was part of a murder-mystery novel, I thought: what could be the cause of this disparity? To be a good detective, any explanation, no matter how implausible, must be thought of and reasoned out.

One explanation of the different angles is that, somehow, the speed of light changed in the room. This is the same principle behind Snell’s Law, which explains the refraction of light as it travels between air and water. Since the speed of light in air ( $c_1$ ) is different than the speed of light in water ( $c_2$ ), the angle of incidence ( $\theta_1)$ is different from the angle of refraction ( $\theta_2$ ), but they are related through the formula

$\displaystyle \frac{\sin \theta_1}{c_1} = \frac{\sin \theta_2}{c_2}$ .

This relationship occurs because of Fermat’s principle, which says that light always travels in a path that requires the least amount of time. Ordinarily, this means that light travels in a straight line. However, if the speed of light should change (say, when traveling through both air and water), then the path of the light is refracted.

Fermat’s principle also explains why light reflects at equal angles if the speed of light is constant (as amusingly illustrated in this PBS video by Dianna Cowern, a.k.a. Physics Girl). However, if the speed of light should somehow change in the room at the point where the light reflects, then the light would bounce at a different angle for the same reason that Snell’s Law works.

In this case, the angles $\theta_1$ and $\theta_2$ are complementary to the 52-degree and 72-degree angles, respectively. By the cofunction trigonometric identities, this means that

$\sin \theta_1 = \cos 52^\circ \quad$ and $\quad \sin \theta_2 = \cos 72^\circ$ ,

so that Snell’s Law can be rewritten as

$\displaystyle \frac{c_1}{c_2} = \frac{\cos 52^\circ}{\cos 72^\circ} \approx 1.992$ .

In other words, one explanation for the unusual path of light is that the speed of light was almost exactly twice as fast in one part of room than in the other part… and the exact threshold of this change occurred at the point where the light hit the mirror. Perhaps there was some kind of fog, mist, or other contaminant in the air near poor Miss Dawson that was so thick that light slowed to half its usual speed. So that’s one explanation.

The other explanation, of course, is that the artist who drew the picture just did a lousy job depicting the reflected light.

As this was part of a murder-mystery, both options are still open to investigation. (Yes, that was tongue-in-cheek.)

For what it’s worth, the figure in my book was not exactly the same as Sayers’ original drawing — clearly, modern word processing was used that was unavailable in the 1930s. One of these days, I may visit the Wade Center in Wheaton, Illinois, which has an impressive collection of Sayers’ works, to peruse a first-run printing of Unnatural Death to see if the figure in my book is faithful to the one that appeared when the novel was first published.

An algebra and trigonometry–based proof of Kepler’s First Law

The proofs of Kepler’s Three Laws are usually included in textbooks for multivariable calculus. So I was very intrigued when I saw, in the Media Reviews of College Mathematics Journal, that somebody had published a proof of Kepler’s First Law that only uses algebra and trigonometry. Let me quote from the review:

Kepler’s first law states that bounded planetary orbits are elliptical. This law is presented in introductory textbooks, but the proof typically requires intricate integrals or vector analysis involving an accidental degeneracy. Simha offers an elementary proof of Kepler’s first law using algebra and trigonometry at the high school level.
https://doi.org/10.1080/07468342.2022.2026089

Once upon a time, I taught Precalculus for precocious high school students. I wish I had known of this result back then, as it would have been a wonderful capstone to their studies of trigonometry and the conic sections.

The preprint of this result can be found on arXiv. (The proof only addresses Kepler’s First Law and not the Second and Third Laws.) The actual article, for those with institutional access, was published in American Journal of Physics Vol. 89 No. 11 (2021): 1009-1011.

Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Morgan Mayfield. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent.

How could you as a teacher create an activity or project that involves your topic? I want to provide some variety for opportunities to make this an engaging opportunity for Precalculus students and some Calculus students. Here are my three thoughts: IDEA 1: For precalculus students in a regular or advanced class, have them derive this formula in groups. After students are familiar with the Pythagorean identities and with angle sum identities, group students and ask them to derive a formula for double angles Sin(2θ), Cos(2θ), Tan(2θ). Let them struggle a bit, and if needed give them some hints such as useful formulas and ways to represent multiplication so that it looks like other operations. From here, encourage students to simplify when they can and challenge students to find the other formulas of Cos(2θ). Ask students to speculate instances when each formula for Cos(2θ) would be advantageous. This gives students confidence in their own abilities and show how math is interconnected and not just a bunch of trivial formulas. Lastly, to challenge students, have them come up with an alternative way to prove Tan(2θ), notably Sin(2θ)/Cos(2θ). This would make an appropriate activity for students while having them continue practicing proving trigonometric identities. IDEA 2: This next idea should be implemented for an advanced Precal class, and only when there is some time to spare. Euler was an intelligent man and left us with the Euler’s Formula: $e^{ix}=\cos x + i \sin x$ . Have Precalculus students suspend their questions about where it comes from and what it is used for. This is not something they would use in their class. Reassure them that for what they will do, all they need to understand is imaginary numbers, multiplying imaginary numbers, and laws of exponents. Have them plug in x = A + B and simplify the right-hand side of the equation so that we get: $\cos(A+B)+i\sin(A+B)= a + bi$ where $a$ and $b$ are two real numbers. The goal here is to get $\cos(A+B)+i\sin(A+B)= \cos \theta \cos \theta - \sin \theta \sin \theta + (\sin \theta \cos \theta + \cos \theta \sin \theta)i$ . All the steps to get to this point is Algebra, nothing out of their grasp. Now, the next part is to really get their brains going about what meaning we can make of this. If they are struggling, have them think about the implications of two imaginary numbers being equal; the coefficient of the real parts and imaginary parts must be equal to each other. Lastly, ask them if these equations seem familiar, where are they from, and what are they called…the angle sum formulas. From here, this can lead into what if x=2A? Students will either brute force the formula again, and others will realize x = A + A and plug it in to the equation they just derived and simplify. This idea is a 2-in-1 steal for the angle sum formulas and double angle formulas. It’s biggest downside is this is for Sin(2θ) and Cos(2θ). IDEA 3: Take IDEA 2, and put it in a Calculus 2 class. Everything that the precalculus class remains, but now have the paired students prove the Euler’s Formula using Taylor Series. Guide them through using the Taylor Series to figure out a Taylor Series representation of $e^x$ , $sin x$ , and $cos x$ . Then ask students to find an expanded Taylor Series of to 12 terms with ellipses, no need to evaluate each term, just the precise term. Give hints such as $i^2= -1$ and to consider $i^3=i^2 \cdot i = -i$ and other similar cases. Lastly, ask students to separate the extended series in a way that mimics $a + bi$ using ellipses to shows the series goes to infinity. What they should find is something like this:

Look familiar? Well it is the addition of two Taylor Series that represent Sin(x) and Cos(x). This is the last connection students need to make. Give hints to look through their notes to see why the “a” and “b” in the imaginary number look so familiar. This, is just one way to prove Euler’s Formula, then you can continue with IDEA 2 until your students prove the angle sum formulas and double angle formulas.

How does this topic extend what your students should have learned in previous courses? Students in Texas will typically be exposed to the Pythagorean Theorem in 8^th grade. At this stage, students use $a^2+b^2=c^2$ to find a missing side length. Students may also be exposed to Pythagorean triples at this stage. Then at the Geometry level or in a Trigonometry section, students will be exposed to the Pythagorean Identity. The Identity is $\sin^2 \theta + \cos^2 \theta = 1$ . I think that this is not fair for students to just learn this identity without connecting it to the Pythagorean Theorem. I think it would be a nice challenge student to solve for this identity by using a right triangle with hypotenuse c so that Sin (θ) = b/c and cos (θ) = a/c, one could then show either $c^2 \sin^2 \theta + c^2 \cos^2 \theta = c^2$ and thus $c^2(\sin^2 \theta + \cos^2 \theta) = c^2$ or one could show $(a/c)^2 + (b/c)^2 = (c/c)^2 = 1$ (using the Pythagorean theorem). From here, students learn about the angle addition and subtraction formulas in Precalculus. This is all that they need to derive the double angle formulas.

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

$\tan(\alpha + \beta) = \displaystyle \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

$\tan(\alpha - \beta) = \displaystyle \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

This would be a good challenge exercise for students to do in pairs. Sin(2θ) = Sin(θ + θ), Cos(2 θ) = Cos(θ + θ), Tan(2θ) = Tan(θ + θ). Now we can apply the angle sum formula where both angles are equal: Sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ) Cos(2θ) = cos(θ)cos(θ) – sin(θ)sin(θ) = (We use a Pythagorean Identity here) Tan(2θ) = $\displaystyle \frac{\tan \theta + \tan \theta}{1 - \tan^2 \theta} = \frac{2 \tan \theta}{1-\tan^2 \theta}$ Bonus challenge, use Sin(2θ) and Cos(2θ) to get Tan(2θ). Well, if $\tan \theta = \displaystyle \frac{\sin \theta}{\cos \theta}$ , then

$\tan 2\theta = \displaystyle \frac{\sin 2\theta}{\cos 2\theta}$

$= \displaystyle \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}$

$= \displaystyle \frac{ \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} }{ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} }$

$= \displaystyle \frac{2 \tan \theta}{1 - \tan^2 \theta}$

The derivations are straight forward, and I believe that many students get off the hook by not being exposed to deriving many trigonometric identities and taking them as facts. This is in the grasp of an average 10^th to 12^th grader.

What are the contributions of various cultures to this topic? I have included four links that talk about the history of Trigonometry. It seemed that ancient societies would need to know about the Pythagorean Identities and the angles sum formulas to know the double angle formulas. Here is our problem, it’s hard to know who “did it first?” and when “did they know it?”. Mathematical proofs and history were not kept as neatly written record but as oral traditions, entertainment, hobbies, and professions. The truth is that from my reading, many cultures understood the double angle formula to some extent independently of each other, even if there was no formal proof or record of it. Looking back at my answer to B2, it seems that the double angle formula is almost like a corollary to knowing the angle sum formulas, and thus to understand one could imply knowledge of the other. Perhaps, it was just not deemed important to put the double angle formula into a category of its own. Many of the people who figured out these identities were doing it because they were astronomers, navigators, or carpenters (construction). Triangles and circles are very important to these professions. Knowledge of the angle sum formula was known in Ancient China, Ancient India, Egypt, Greece (originally in the form of broken chords theorem by Archimedes), and the wider “Medieval Islamic World”. Do note that that Egypt, Greece, and the Medieval Islamic World were heavily intertwined as being on the east side of the Mediterranean and being important centers of knowledge (i.e. Library of Alexandria.) Here is the thing, their knowledge was not always demonstrated in the same way as we know it today. Some cultures did have functions similar to the modern trigonometric functions today, and an Indian mathematician, Mādhava of Sangamagrāma, figured out the Taylor Series approximations of those functions in the 1400’s. Greece and China for example relayed heavily on displaying knowledge of trigonometry in ideas of the length of lines (rods) as manifestations of variables and numbers. Ancient peoples didn’t have calculators, and they may have defined trigonometric functions in a way that would be correct such as the “law of sines” or a “Taylor series”, but still relied on physical “sine tables” to find a numerical representation of sine to n numbers after the decimal point. How we think of Geometry and Trigonometry today may have come from Descartes’ invention of the Cartesian plane as a convenient way to bridge Algebra and Geometry. References: https://www.mathpages.com/home/kmath205/kmath205.htm https://en.wikipedia.org/wiki/History_of_trigonometry https://www.ima.umn.edu/press-room/mumford-and-pythagoras-theorem

Engaging students: Computing trigonometric functions using a unit circle

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 Alizee Garcia. Her topic, from Precalculus: computing trigonometric functions using a unit circle.

How can this topic be used in your students’ future courses in mathematics or science?

Being able to compute trig functions using a unit circle will be the base of knowledge for all further calculus classes, as well as others. Being able to understand and use a unit circle will also allow students to start to memorize the trigonometric functions. One of the most important things from pre-calculus to all other calculus classes was being able to solve trig functions and having the unit circle memorized was very useful. Although there are trig functions and values outside of the unit circle, the unit circle almost is like the foundation for trigonometry. Most, if not all, calculus classes after pre-calculus will expect students to have the unit circle memorized. Although it can be solved using a calculator, this will allow equations and problems to be solves easier with less thought when a student knows the unit circle. Even outside of calculus classes, the unit circle is one of many important aspects in math classes.

How does this topic extend what your students should have learned in previous courses?

Before students learn how to compute trigonometric functions using a unit circle, they learn about the trig functions by themselves. This usually starts in high school geometry where students learn sine, cosine, and tangent, yet they do not use them in the way a unit circle does. Most schools only teach the students how to use the calculator to compute the functions to solve sides or angles for triangles. As students enter pre-calculus, they use what they have learned about the trig functions in order to apply them to the unit circle. This will allow students to see that using trig functions can still be used to solve triangles, but it can also be used to solve many other things. Once they learn the unit circle, they will see more examples in which they will apply the functions and make connections to real-world scenarios that they can also be applied to.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are probably many simulations and websites that can help students compute trig functions using the unit circle, but I think something that will engage the students is a Kahoot or Quizziz that will help the students memorize the unit circle. Giving students an opportunity to apply what they learned into a friendly competition not only gives them practice but will also let them be engaged. Other technology resources such as videos or a website that is teaching the lesson does not really allow the students to apply what they know rather than just being lectured. Although some websites and technology can be useful, I personally, enjoy giving students the opportunity to work out problems as well as being engaged. Also, using calculators could be helpful to check answers but if they have a unit circle it might not be necessary unless they do not have the unit circle in front of them.

My Favorite One-Liners: Part 121

I’ll use this one-liner when I ask my students to do something that’s a little conventional but nevertheless within their grasp. For example, consider the following calculation using a half-angle trigonometric identity:

$\cos \displaystyle \frac{5\pi}{8} = \cos \displaystyle \left( \frac{1}{2} \cdot \frac{5\pi}{4} \right)$

$= \displaystyle - \sqrt{ \frac{1 + \cos 5\pi/4}{2} }$

$= \displaystyle - \sqrt{ \frac{ 1 - \displaystyle \frac{\sqrt{2}}{2}}{2} }$

$= \displaystyle - \sqrt{ \frac{ ~~~ \displaystyle \frac{2-\sqrt{2}}{2} ~~~}{2} }$

$= \displaystyle - \sqrt{ \frac{2 - \sqrt{2}}{4}}$

$= \displaystyle - \frac{ \sqrt{2 - \sqrt{2}}}{\sqrt{4}}$

$= \displaystyle - \frac{ \sqrt{2 - \sqrt{2}}}{2}$

That’s certainly a very complicated calculation, with plenty of predictable places where a student might make an inadvertent mistake.

In my experience, one somewhat surprising place that can trip up students seeing such a calculation for the first time is the very first step: changing $\displaystyle \frac{5\pi}{8}$ into $\displaystyle \frac{1}{2} \cdot \frac{5\pi}{4}$ . Upon reflection, perhaps this isn’t so surprising: students are very accustomed to taking a complicated expression like $\displaystyle \frac{1}{2} \cdot \frac{5\pi}{4}$ and making it simpler. However, they aren’t often asked to take a simple expression like $\displaystyle \frac{5\pi}{8}$ and make it more complicated.

So I try to make this explicitly clear to my students. A lot of times, we want to make a complicated expression simple. Sometimes, we have to go the other direction and make a simple expression more complicated. Students should be able to do both. And, to try to make this memorable for my students, I use my one-liner:

“In the words of the great philosopher, you gotta know when to hold ’em and know when to fold ’em.”

Yes, that’s an old song reference. My experience is that most students have heard the line before but unfortunately can’t identify the singer: the late, great Kenny Rogers.

Engaging students: Using right-triangle trigonometry

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 Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry.

A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation.

A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket.

D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared ( $a^2+b^2=c^2$ ). This actually was a special case for the law of cosines ( $c^2=a^2+b^2-2ab\cos(\theta)$ ). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, $\sin^2 x+\cos^2 x=1$ . This can be seen in the unit circle where the legs of the right triangle are $\sin x$ and $\cos x$ and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.

My Favorite One-Liners: Part 120

I used these shirts as props when teaching Precalculus this week, and they worked like a charm.

After deriving the three Pythagorean identities from trigonometry, I told my class that I got these hand-made his-and-hers T-shirts for my wife’s birthday a couple of years ago. If you can’t see from the picture, one says $\sin^2 \theta$ and the other $\cos^2 \theta$ .

After holding up the shirts, I then asked the class what mathematical message was being communicated.

After a few seconds, someone ventured a guess: “We add up to 1?”

I answered, “That’s right. Together, we’re one.”

Whereupon the class spontaneously reacted with a loud “Awwwwwwwwww.”

I was exceedingly happy.

Engaging students: Using right-triangle trigonometry

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 Cameron Story. His topic, from Precalculus: using right-triangle trigonometry.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Most right-angle trigonometry word problems involve giving two measurements of a triangle (angle, sides or both) and asking the students to solve for the missing piece. I argue that these problems are fine for practice, but one has to admit these problems encourage “plugging and chugging” along with their formula sheets.

To make things interesting, I would use something along the lines of this word problem from purplemath.com:

“You use a transit to measure the angle of the sun in the sky; the sun fills 34′ of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest mile,” (Stapel, 2018).

This is incredible! Using trigonometry, students can find out the diameter of the entire sun just by knowing how far away it is and how much of the sky the sun takes up. If you were to use this word problem in a experimental type of project, I strongly recommend using the moon for measurement instead; you can probably guess why measuring the sun in the sky is a BAD idea.

What are the contributions of various cultures to this topic?

One amazing culture to contribute to the study of triangles and trigonometry were the Ancient Babylonians, who lived in what is now Iraq about 4,000 years ago. Archaeologists have found clay tablets from 1800 BC where the Babylonians carved and recorded various formulas and geometric properties. There were several such tablets found to have been lists of Pythagorean triples, which are integer solutions to the famous equation $a^2+b^2=c^2$ .

The Greeks, while going through their own philosophical and mathematical renaissance, gave the namesake for trigonometry. Melanie Palen, writer for the blog Owlcation, makes is very clear why trigonometry “… sounds triangle-y.” The word trigonometry is derived from two Greek words – ‘trigonon’ which means ‘triangle’ and ‘metron’ meaning ‘measure.’ “Put together, the words mean “triangle measuring”” (Palen, 2018).

How can technology (YouTube) be used to effectively engage students with this topic?

In the YouTube video “Tattoos on Math” by the YouTube channel 3Blue1Brown (link: https://youtu.be/IxNb1WG_Ido), Grant Sanderson offers a unique perspective on the six main trigonometric functions. In the video. Grant explains how his friend Cam has the initials CSC, which is how we notationaly represent the cosecant function. Not only is this engaging because most students wouldn’t even think of seeing tattoos in math class, but also because Grant always backs up the mathematical content in his videos with beautiful animations.

Students know how sine and cosine functions are represented geometrically; these are just the “legs” of a right-angled triangle. Most students, however, only see the other four trigonometric functions as formulas to be solved. However, as Grant cleverly explains and visualizes in this video, all of these functions have geometric representations as well when paired with the unit circle. This video (moreover, this entire YouTube channel) can be helpful to those visual-learning students who need more than a formula to be convinced of something like the cosecant function.

References:

3Blue1Brown YouTube Video: https://youtu.be/IxNb1WG_Ido

Palen, Melanie. “What Is Trigonometry? Description & History of Trig.” Owlcation, Owlcation, 25 July 2018, owlcation.com/stem/What-is-Trigonometry.

Stapel, Elizabeth. “Right-Triangle Word Problems.” Purplemath, 2018, http://www.purplemath.com/modules/rghtprob.html