Xjyuk

I have a question regarding tutoring a calculus student. They need to prove trig identities such as $$frac11-sin x+frac11+sin x=2sec^2x.$$ Doing this kind of problem is very tedious and time consuming. Is it really so necessary to focus on this for the learning of calculus or can this part be done more superficially?

The specific identity
beginequationtagA
tfrac11 - sinx + tfrac11 + sinx = 2sec^2x
endequation
as such is probably not often encountered, but simplifications akin to beginequationtagB
tfrac11-t + tfrac11 + t = tfrac21 - t^2
endequation
occur frequently. For example, integration via partial fractions requires undoing such a simplification, and this manipulation is impossible to understand for someone who does not understand the forward operation being undone. In the same spirit, seeing the formal similarity between (A) and (B) is relevant when it comes to making changes of variables in integrals.

Understanding that the identity (A) on the one hand involves the general algebraic identity (B) and on the other hand uses the trigonometric identity $sin^2x + cos^2x = 1$, and understanding how to separate these two statements, is useful for developing the sort of calculational skills that are generally necessary for making progress in calculus.

In more mercantile terms, experience teaching calculus suggests that students who cannot make manipulations such as (A) are unlikely to pass a university calculus course.

Finally, characterizing (A) as very tedious and time consuming seems to me simply wrong, as it is neither.

Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger picture in calculus?" has been on my mind lately.

Here's where my thoughts have fleshed out in regards to trig so far.

• Periodic behavior is widespread throughout science, engineering, and the humanities, and sinusoidal functions are typically used to model periodic behavior. So simply ignoring trig is not an option.


• There is absolutely prior knowledge that students should have about trig that will be essential to understand key ideas. For instance, you cannot understand the derivative of sine without knowing that $sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha$. You're not going to understand how to calculate $intfracdxsqrt1+x^2$ without understanding inverse trig functions.


• Beyond that, there really is a lot of fluff that doesn't seem to serve any real-world purpose. I'm frankly on the fence about whether secant, cosecant, and cotangent should largely go the way of versine and exsecant. Esoteric identities like the one the OP posts are at best useful to facilitate hand calculations based on a relatively small number of trig tables, which is not a 21st century concern.

In the end, of course, a student needs to know enough trig identities to be prepared for their final exam, and trig has enormous utility in the real world. But I am sympathetic to the argument that the twentieth century advanced algebra curriculum is not wholly authentic preparation for twenty-first century calculus.

From a historical perspective, knowing these identities used to be somewhat more important than now. Prior to the invention of logarithms, people who needed to do lots of sophisticated calculations [esp. astronomers] resorted to a technique know as 'prostapharesis'. This involved combining certain trigonometric identities to produce equations (e.g) having a product of trig functions on one side and a sum or difference of trig functions on the other. This allowed people to transform a multiplication into an addition or subtraction, like logarithms do (but in a somewhat more cumbersome manner).

1. Algebraic skills to do manipulations in general are important. And this one is not that hard. Good practice. Get dirty and do it.




2. There's some applications when you get to trig subs of quadratic radicals and the like.

Once the student figures out the process - start from one side of what is to be proven, use known formulas to get the other side, done - the rest is getting intuition about which formula to use in the process and when. I guess this intuition might be useful if they need to evaluate tricky integrals analytically, or if they do something very trigonometric later in their life.

If getting good scores from exams is important to the student, then the importance of trigonometry depends on the instructor and the syllabus. We are probably ill advised to make guesses on those, especially since there is no country specified.

For what it is worth, I have never been taught or needed any trigonometric functions but sin, cos, tan and their inverses.

There are many calculus textbooks that use no trig. They may be called "Calculus for Business" or "For Biology" or "For Social Science".

random example:

Of course math, physical science, and engineering, definitely use parts of calculus connected with trig functions. I would have thought that Business would be interested in cyclic phenomena, but what do I know?

Trigonometric substitutions are useful for solving many integrals in closed form and learning how to solve integrals is a major part of most calculus courses. Often more than half of university-level "Calculus II" is concerned with integration techniques. Without trigonometric identities, it may not be obvious how to solve $int left( frac11- sin x + frac11+sin x right)~mathrmdx$. However, $int sec^2 x~mathrmdx$ is included in many integration tables and happens to be simply $tan x + C$.