This is part of the “Memorize or Melt” series, for more information go here if you haven’t read the introductory post on this topic.

For non-US readers, the following is “syntactic sugar” typically used in the US.

• \(\sec(x) = \frac{1}{\cos(x)}\) where \(\sec\) is “secant”
• \(\csc(x) = \frac{1}{\sin(x)}\) where \(\csc\) is “cosecant”
• \(\cot(x) = \frac{1}{\tan(x)}\) where \(\cot\) is “cotangent”

At some point, and it is not consistent, students are exposed to “Trigonometry” — that subject that deals with sines, cosines, tangents, triangles, and maybe sometimes circles depending on how it is taught. The bane of many students are the trig identities. For some reason or another there is a mindset among some instructors that these identities have to be memorized. To be fair, there are plenty of instructors who do not at all require memorization of trig identities. Also, there is a bit of awkwardness as to when students take trigonometry. Some take Trigonometry after a Geometry course but before pre-Calculus (a hodgepodge course in which students typically learn about functional notation, exponentials, logarithms, complex numbers, matrices, etc.). Others take Trigonometry concurrently with Pre-Calculus. Others have Trigonometry has part of their pre-Calculus course. And still others have Trigonometry as part of their Calculus course. So given the variety in a student’s math background, the teaching style tends to vary out of necessity as well; and when students have not had enough exposure to some foundational mathematics coursework, teaching a Trigonometry course can be a daunting challenge.

This author’s opinion is that Trigonometry is best taught as part of a pre-Calculus course — though whether a pre-Calculus course should even exist is a different matter for another post.

One of the first identities that students are taught is $$\sin^{2}(x) + \cos^{2}(x) = 1$$ but then from here, students are often instructed to commit to memory the following:

• \(\tan^{2}(x) + 1 = \sec^{2}(x)\)
• \(1 + \cot^{2}(x) = \csc^{2}(x)\)

Then there is another list of identities — the double- and half-angle identities. Here are a few:

• \(\cos(2x) = \cos^{2}(x) – \sin^{2}(x) = 2\cos^{2}(x)-1 = 1-2\sin^{2}(x)\)
• \(\sin(2x) = 2\sin(x)\cos(x)\)
• \(\cos(\frac{1}{2}x) = \pm\sqrt{\frac{1+\cos(x)}{2}}\)
• \(\sin(\frac{1}{2}x) = \pm\sqrt{\frac{1-\cos(x)}{2}}\)

Of course, the list goes on. There are addition identities, product-to-sum identities, sum-to-product identities, etc. So the first, obvious question is, why have students memorize / “learn” these identities in the first place? What is the point? Here are two typical “teaching points”:

• “Suppose we wanted to know the value of \(\sin(\frac{\pi}{8})\). We could recognize that \(\frac{\pi}{8}\) is just half of \(\frac{\pi}{4}\) and use the half-angle formula to answer this. In this way, we can find the value of trigonometric functions at specific angles based on the value of trigonometric functions at other angles.”
• “These are nice relationships that can help to simplify problems. For example, in Calculus, certain integrals can be easily evaluated using identities.”

The first point, in this author’s opinion, is just an antiquated notion. Maybe in the days of slide rules, these identities were useful. But today, with the abundance of calculators what possible reason should a student have to learn these? In fact, a statement like the first one — and I invite readers to do a simple internet query of “why are trig identities useful to learn” and see what is said — is often countered by students who say, exactly, “Why can’t I just plug this into my calculator?” At this moment, what is an instructor to say? There is nothing an instructor can say. It is an absolutely correct question for a student to ask and there is no good answer an instructor can give other than “You can just plug it into your calculator.” And at this moment, another student has begun to lose interest.

The second point has a bit more validity even if it is a bit self-propagating (“We are learning these identities because they can be used for solving other problems which we are learning to solve so that we can solve other problems …”). If the instructor simply wants to provide the identities as a reference to help students build a comfort level with the mechanics of working with sines, cosines, and tangents, then that’s not so bad. If, however, the instructor feels that the student is better off memorizing identities because it will be helpful later on, then this is a disservice to the students.

There is certainly a geometric meaning to the double- and half-angle formulas and that can be explained to students. But what about triple-angle identities? Why aren’t students asked to learn those? There must be problems that could be simplified using this identity
$$\sin(3x) = 3\cos^{2}(x)\sin(x) – \sin^{3}(x)$$
or this quintuple-angle identity
$$\cos(5x) = \cos^{5}(x) – 10\cos^{3}(x)\sin^{2}(x) + 5\cos(x)\sin^{4}(x)$$

The answer is probably what was said in a previous post, that there is just a feedback loop. Students learn a certain set of material now because previous students had learned that material.

The world has changed dramatically in the last several decades and the continued presence of things like trigonometric identities as things to learn because of their calculation benefits are signs that mathematics education hasn’t kept pace with technology.

A few suggestions (and these will become blog topics at some point):

• Instructors should find more meaningful exercises to give to students rather than the same tired questions like “Evaluate \(\sin(\frac{\pi}{8})\).”
• Instructors can spend some time with students exploring how calculators evaluate trigonometric functions (e.g., Taylor series). Though, understandably, Taylor series requires some Calculus knowledge, the instructor can, nevertheless, demonstrate how trigonometric functions can be constructed from “infinite-degree polynomials.” (The author recognizes that polynomials are by definition finite in degree.)
• If complex numbers haven’t been introduced, then this is a good time to digress and introduce Euler’s formula \(e^{ix} = \cos(x) + i\sin(x)\). Then, at least, the instructor can have students derive, algebraically, any \(n\)-angle identity (where \(n\) is an integer). All the other trig identities can be derived as well from Euler’s formula. This also helps to introduce or reinforce the binomial theorem as well as keep some mechanical skills fresh.
• Instructors can give a high-level overview of Fourier series and their use in signal processing.

The message again is, buck the trend a little bit and save your students. There is so much more that we can do in a trigonometry course. Use their access to and desire to use calculators and other computational tools as educational tools — these devices exist and as educators we have to learn to work with and integrate them into our instructional methodology.