Trigonometry pun

Source: https://www.facebook.com/743378802479548/photos/a.743768985773863.1073741828.743378802479548/792810117536416/?type=3&theater

Engaging students: Graphing the 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 Jessica Bonney. Her topic, from Precalculus: graphing the sine and cosine functions.

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

 

A fun activity for students to learn how to graph the sine and cosine function would be having them build the graph using spaghetti and yarn. Students would start out with a simple warm-up to help them recall the different values of sine and cosine on the unit circle depending on the given angle. After the warm-up, I would then pair students off into groups of two and have them create the graphs, one creating the sine graph and the other creating the cosine graph. The first step in this activity would be for students to take their yarn and wrap it around the unit circle, marking each significant angle on the yarn with a marker. Next, students will create the x and y-axis on their paper, making the x-axis along the center of the paper (labeling it Θ) and the y-axis about 1/3 of the way from the left-end of the paper (labeling it either cosΘ or sinΘ). They then lay the yarn on the x-axis, with the end on the origin, which represents 0 radians, and using the marks they made on the yarn they will mark and label each point on the x-axis. Going back to the unit circle, students will then measure the major angles of either sine or cosine with spaghetti. This part is used to help solidify their understanding that the values of x and y correspond to cosine and sine. After measuring and cutting the spaghetti, students will then glue the spaghetti down to the matching angle on the coordinate plane. Once they have finished gluing their pasta down, students will take a marker and draw the curve. To end the lesson, I would have the students do a think-pair-share, answering the following question: Why is the function curve wider that the unit circle? After, I would have students compare their graphs and demonstrate how they found their graph.

 

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

 

Graphing the sine and cosine functions is a topic that students will carry on with them throughout the rest of their future science and mathematics courses. For starters, they will need to know how to do this for all advanced calculus or trigonometry classes they will take in high school or even in college. An example of this would be, when the students learn how to derive the tangent, cotangent, secant, and cosecant functions and graphs. Next, students will use this more in depth in their future physics courses. They will be able to relate waveforms and vibrations to that of specific sine and cosine graphs. Vibrations are graphs with the equations y=sin(t) or y=cos(t), and the time needed for one oscillation across the x-axis is referred to as a period. Waveforms are graphs with the equations y=sin(x) or y=cos(x), and the distance needed for one oscillation across the x-axis is referred to as a wavelength. As you can see, this particular topic in pre-calculus is an important piece in laying the foundation in their future academics and beyond.

 

 

 

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

 

            For starters, the word trigonometry comes from the Greek word trignon, meaning “triangle”, and metron, meaning “to measure.” Before the 16th century, trigonometry was mainly used for computing the unaccounted for parts of a triangle when the other parts were given. When it comes to ancient civilizations, Egyptians had a collection of 84 algebra, arithmetic, and geometry problems called the Rhind Papyrus. This showed that the Egyptians had some knowledge about the triangle, almost like a “pre-trigonometry”. It wasn’t until the Greeks, that trigonometry began to make sense. Hipparchus was the first to construct a table of the values of trigonometric functions. The next key contribution to trigonometry as we know it came from India. The author of the Aryabhatiya used words for “chord” and “half-chord” which was later shortend to jya or jiva. Following this, Muslim scholars translated the words into Arabic, which was then translated into Latin. An English minister, Edmund Gunter, first used the shortened term that we know, sin, in 1624. In 1614, John Napier invented logarithms, the final major contribution of classical trigonometry.

 

References:

 

https://www.britannica.com/topic/trigonometry

 

http://betterlesson.com/lesson/437440/graphs-of-sine-and-cosine

 

http://www.algebralab.org/lessons/lesson.aspx?file=trigonometry_trigperiodfreq.xml

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

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How does this topic extend what your students should have already learned?

A major factor that simplifies deriving the double angle formulas is recalling the trigonometric identities that help students “skip steps.” This is true especially for the Sum formulas, so a brief review of these formulas in any fashion would help students possibly derive the equations on their own in some cases. Listed below are the formulas that can lead directly to the double angle formulas.

A list of the formulas that students can benefit from recalling:

  • Sum Formulas:
    • sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
    • cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
    • tan(a+b) = [tan(a) +tan(b)] / [1-tan(a)tan(b)]

 

  • Pythagorean Identity:
    • Sin2 (a) + Cos2(a) = 1

 

This leads to the next topic, an activity for students to attempt the equation on their own.

 

 

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

I’m a firm believer that the more often a student can learn something of their own accord, the better off they are. Providing the skeletal structure of the proofs for the double angle formulas of sine, cosine, and tangent might be enough to help students reach the formulas themselves. The major benefit of this is that, even though these are simple proofs, they have a lot of variance on how they may be presented to students and how “hands on” the activity can be.

I have an example worksheet demonstrating this with the first two double angle formulas attached below. This is in extremely hands on format that can be given to students with the formulas needed in the top right corner and the general position where these should be inserted. If needed the instructor could take this a step further and have the different Pythagorean Identities already listed out (I.e. Cos2(a) = 1 – Sin2(a), Sin2(a) = 1 – Cos2(a)) to emphasize that different formats could be needed. This is an extreme that wouldn’t take students any time to reach the conclusions desired. Of course a lot of this information could be dropped to increase the effort needed to reach the conclusion.

A major benefit with this also is that even though they’re simple, students will still feel extremely rewarded from succeeding on this paper on their own, and thus would be more intrinsically motivated towards learning trig identities.

 

 

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How can Technology be used to effectively engage students with this topic?

When it comes to technology in the classroom, I tend to lean more on the careful side. I know me as a person/instructor, and I know I can get carried away and make a mess of things because there was so much excitement over a new toy to play with. I also know that the technology can often detract from the actual math itself, but when it comes to trigonometry, and basically any form of geometric mathematics, it’s absolutely necessary to have a visual aid, and this is where technology excels.

The Wolfram Company has provided hundreds of widgets for this exact purpose, and below, you’ll find one attached that demonstrates that sin(2a) appears to be equal to its identity 2cos(a)sin(a). This is clearly not a rigorous proof, but it will help students visualize how these formulas interact with each other and how they may be similar. The fact that it isn’t rigorous may even convince students to try to debunk it. If you can make a student just irritated enough that they spend a few minutes trying to find a way to show you that you’re wrong, then you’ve done your job in that you’ve convinced them to try mathematics for a purpose.

After all, at the end of the day, it doesn’t matter how you begin your classroom, or how you engage your students, what matters is that they are engaged, and are willing to learn.

Wolfram does have a free cdf reader for its demonstrations on this website: http://demonstrations.wolfram.com/AVisualProofOfTheDoubleAngleFormulaForSine/

 

References

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.

sine1

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.

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

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

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

TIarcsin45

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

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

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

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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.