Engaging students: Verifying trigonometric identities

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 Tracy Leeper. Her topic, from Precalculus: verifying trigonometric identities.

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Many students when first learning about trigonometric identities want to move terms across the equal sign, since that is what they have been taught to do since algebra, however, in proving a trigonometric identity only one side of the equality is worked at a time. Therefore my idea for an activity to help students is to have them look at the identities as a puzzle that needs to be solved. I would provide them with a basic mat divided into two columns with an equal sign printed between the columns, and give them trig identities written out in a variety of forms, such as \sin^2 \theta + \cos^2 \theta on one strip, and 1 written on another strip. Other examples would also include having \tan^2 \theta on one, and \sin^2 \theta/\cos^2 \theta on another. The students will have to work within one column, and step by step, change one side to eventually reflect the term on the other side, and each strip has to be one possible representation of the same value. By providing the students with the equivalent strips, they will be able to construct the proof of the identity. I feel that giving them the strips will allow them to see different possibilities for how to manipulate the expression, without leaving them feeling lost in the process, and by dividing the mat into columns, they can focus on one side, and see that the equivalency is maintained throughout the proof. The students would need to arrange the strips into the correct order to prove the left hand side is equivalent to the right hand side, while reinforcing the process of not moving anything across the equal sign.

 

 

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Trigonometry identities are used in most of the math courses after pre-calculus, as well as the idea of proving an equivalency. If the students learn the concept of proving an equivalency that will help them construct proofs for any future math courses, as well as learning to look at something given, and be able to see it as parts of a whole, or just be able to write it a different way to assist with the calculations. If students learn to see that

1 = \sin^2 x + \cos^2 x = \sec^2 x - \tan^2 x = \csc^2 x - \cot^2 x,

their ability to manipulate expressions will dramatically improve, and their confidence in their ability will increase, as well as their understanding of the complexities and relations throughout all of mathematics. The trigonometric identities are the fundamental part of the relationships between the trig functions. These are used in science as well, anytime a concept is taught about a wave pattern. Sound waves, light waves, every kind of wave discussed in science are sinusoidal wave. Anytime motion is calculated, trigonometry is brought into the calculations. All students who wish to progress in the study of science or math need to learn basic trigonometric identities and learn how to prove equivalency for the identities. Since proving trigonometric identities is also a practice in logical reasoning, it will also help students learn to think critically, and learn to defend their conjectures, which is a valuable skill no matter what discipline the student pursues.

 

 

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For learning how to verify trigonometric identities, I like the Professor Rob Bob (Mr. Tarroy’s) videos found on youtube. He’s very energetic, and very thorough in explaining what needs to be done for each identity. He also gives examples for all of the different types of identities that are used. He is very specific about using the proper terms, and he makes sure to point out multiple times that this is an identity, not an equation, so terms cannot be transferred across the equal sign. He also presents options to use for a variety of cases, and that sometimes things don’t work out, but it’s okay, because you can just erase it and start again. I also like that he uses different colored chalk to show the changes that are being made. He is very articulate, and explains things very well, and makes sure to point out that he is providing examples, but it’s important to remember that there are many different ways to prove the identity presented. I enjoyed watching him teach, and I think the students would enjoy his energy as well.

 

Engaging students: Verifying trigonometric identities

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 Michelle McKay. Her topic, from Precalculus: verifying trigonometric identities.

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

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Engaging students with trigonometric identities may seem daunting, but I believe the key to success for this unit lies within allowing students to make the discovery of the identities themselves.

For this particular activity, I will focus on some trigonometric identities that can be derived using the Pythagorean Theorem. Before beginning this activity, students must already know about the basic trig functions (sine, cosine, and tangent) along with their corresponding reciprocals (cosecant, secant, and cotangent).

Using this diagram (or a similar one), have students write out the relationship between all sides using the Pythagorean Theorem.
Students should all come to the conclusion of: x2 + y2 = r2.

For higher leveled students, you may want to remind them of the adage SohCahToa, with emphasis on sine and cosine for this next part. You might ask, “How can we rearrange the above equation into something remotely similar to a trigonometric function?”

Ultimately, we want students to divide each side by r2. This will give us:

Again, SohCahToa. Students, perhaps with some leading questions, should see that we can substitute sine and cosine functions into the above equation, giving us the identity:

cos2θ + sin2θ = 1

From this newly derived identity, students can then go on to find tan2θ + 1 = sec2θand then 1 + cot2θ = csc2θ.

 

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

For engaging the students and encouraging them to play around with identities, I find the Trigonometric Identities Solver by Symbolab to be a fabulous technological supplement. Students can enter in identities that they may need more help understanding and this website will state whether the identity is true or not, and then provide detailed steps on how to derive the identity.
A rather fun activity that may utilize this site is to challenge the students to come up with their own elaborate trigonometric identity.

SymbolLab

Another online tool students can explore is the interactive graph from http://www.intmath.com. In fact, students could also use this right after they derive the identities from the earlier activity. This site does a wonderful job at providing a visual representation of the trigonometric functions’ relationships to one another. It also allows the students to explore the functions using concrete numbers, rather than the general Ө. Although this site only shows the cos2θ + sin2θ = 1identity in action, it would not be difficult for students to plug in the data from this graph to numerically verify the other identities.

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What are the contributions of various cultures to this topic?

 

The beginning of trigonometry began with the intention of keeping track of time and the quickly expanding interest in the study of astronomy. As each civilization inherited old discoveries from their predecessors, they added more to the field of trigonometry to better explain the world around them. The below table is a very brief compilation of some defining moments in trigonometry’s history. It is by no means complete, but was created with the intention to capture the essence of each civilization’s biggest contributions.

 

Civilization People of Interest Contributions
Egyptians
  • Ahmes
– Earliest ideas of angles.- The Egyptian seked was the cotangent of an angle at the base of a building.
Babylonians – Division of the circle into 360 degrees.- Detailed records of moving celestial bodies (which, when mapped out, resembled a sine or cosine curve).- May have had the first table of secants.
Greek
  • Aristarchus
  • Menelaus
  • Hippocharus
  • Ptolemy
– Chords.- Trigonometric proofs presented in a geometric way.- First widely recognized trigonometric table: Corresponding values of arcs and chords.- Equivalent of the half-angle formula.
Indian
  • Aryabhata
  • Bhaskara I
  • Bhaskara II
  • Brahmagupta
  • Madhava
– Sine and cosine series.- Formula for the sine of an acute angle.- Spherical trigonometry.- Defined modern sine, cosine, versine, and inverse sine.
Islamic
  • Muhammad ibn Mūsā al-Khwārizmī
  • Muhammad ibn Jābir al-Harrānī al-Battānī
  •  Abū al-Wafā’ al-Būzjānī
–          – First accurate sine and cosine tables.-          – First table for tangent values.-          – Discovery of reciprocal functions (secant and cosecant).-          – Law of Sines for spherical trigonometry.-          – Angle addition in trigonometric functions.
Germans – “Modern trigonometry” was born by defining trigonometry functions as ratios rather than lengths of lines.

 

It is interesting to note that while the Chinese were making many advances in other fields of mathematics, there was not a large appreciation for trigonometry until long after they approached the study and other civilizations had made significant contributions.

 

 

Sources

  1. http://www.intmath.com/analytic-trigonometry/1-trigonometric-identities.php
  2. http://www.intmath.com/analytic-trigonometry/trig-ratios-interactive.php
  3. http://symbolab.com/solver
  4. http://www.trigonometry-help.net/history-of-trigonometry.php
  5. http://nrich.maths.org/6843&part=
  6. http://www.scribd.com/doc/33216837/The-History-of-Trigonometry-and-of-Trigonometric-Functions-May-Span-Nearly-4
  7. http://www.britannica.com/EBchecked/topic/605281/trigonometry/12231/History-of-trigonometry

Inverse Functions: Arcsecant (Part 29)

We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of y = \sec x satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of y = \sec^{-1} x uses the interval [0,\pi] — or, more precisely, [0,\pi/2) \cup (\pi/2, \pi] to avoid the vertical asymptote at x = \pi/2 — in order to approximately match the range of \cos^{-1} x. However, when I was a student, I distinctly remember that my textbook chose [0,\pi/2) \cup [\pi,3\pi/2) as the range for \sec^{-1} x.

I believe that this definition has fallen out of favor today. However, for the purpose of today’s post, let’s just run with this definition and see what happens. This portion of the graph of y = \sec x is perhaps unorthodox, but it satisfies the horizontal line test so that the inverse function can be defined.

arcsec3

Let’s fast-forward a couple of semesters and use implicit differentiation (see also https://meangreenmath.com/2014/08/08/different-definitions-of-logarithm-part-8/ for how this same logic is used for other inverse functions) to find the derivative of y = \sec^{-1} x:

x = \sec y

\displaystyle \frac{d}{dx} (x) = \displaystyle \frac{d}{dx} (\sec y)

1 = \sec y \tan y \displaystyle \frac{dy}{dx}

\displaystyle \frac{1}{\sec y \tan y} = \displaystyle \frac{dy}{dx}

 At this point, the object is to convert the left-hand side to something involving only x. Clearly, we can replace \sec y with x. As it turns out, the replacement of \tan y is a lot simpler than we saw in yesterday’s post. Once again, we begin with one of the Pythagorean identities:

1 + \tan^2 y = \sec^2 y

\tan^2 y = \sec^2 y - 1

\tan^2 y = x^2 - 1

\tan y = \sqrt{x^2 - 1} \qquad \hbox{or} \tan y = -\sqrt{x^2 - 1}

So which is it, the positive answer or the negative answer? In yesterday’s post, the answer depended on whether x was positive or negative. However, with the current definition of \sec^{-1} x, we know for certain that the answer is the positive one! How can we be certain? The angle y must lie in either the interval [0,\pi/2) or else the interval [\pi,3\pi/2). In either interval, \tan y is positive. So, using this definition of \sec^{-1} x, we can simply say that

\displaystyle \frac{d}{dx} \sec^{-1} x = \displaystyle \frac{1}{x \sqrt{x^2-1}},

and we don’t have to worry about |x| that appeared in yesterday’s post.

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arcsec2Turning to integration, we now have the simple formula

\displaystyle \int \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} x + C

that works whether x is positive or negative. For example, the orange area can now be calculated correctly:

\displaystyle \int_{-2}^{-2\sqrt{3}/3} \frac{dx}{x \sqrt{x^2 -1}} = \sec^{-1} \left( - \displaystyle \frac{2\sqrt{3}}{3} \right) - \sec^{-1} (-2)

= \displaystyle \frac{7\pi}{6} - \frac{4\pi}{3}

= \displaystyle -\frac{\pi}{6}

So, unlike yesterday’s post, this definition of \sec^{-1} x produces a simple integration formula.

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So why isn’t this the standard definition for \sec^{-1} x? I’m afraid the answer is simple: with this definition, the equation

\sec^{-1} x = \cos^{-1} \left( \displaystyle \frac{1}{x} \right)

is no longer correct if x < -1. Indeed, I distinctly remember thinking, back when I was a student taking trigonometry, that the definition of \sec^{-1} x seemed really odd, and it seemed to me that it would be better if it matched that of \cos^{-1} x. Of course, at that time in my mathematical development, it would have been almost hopeless to explain that the range [0,\pi/2) \cup [\pi,3\pi/2) had been chosen to simplify certain integrals from calculus.

So I suppose that The Powers That Be have decided that it’s more important for this identity to hold than to have a simple integration formula for \displaystyle \int \frac{dx}{x \sqrt{x^2 -1}}