What We Learn When We Learn Trigonometry

Many students hate trigonometry, and for understandable reasons.  It is hard for students to figure out what triangles, sines, and cosines have to do with anything else.  So we shouldn’t be too surprised when they push back a little bit on learning it.  While some trigonometry has real-life usage, the most significant benefit to trig is the intellectual skills it teaches.

Let’s start with real-life uses of trigonometry.  As a computer programmer, I’ve used trigonometry a moderate amount.  Once I wrote an iPad app where I measured body angles and positioning while doing exercises.  This required a decent amount of trigonometry to do all of the measurements.  Most people aren’t aware that almost all game image rendering is based on triangles, so knowing trigonometry will help get into game programming or any visual effects area.

Classical physics uses trigonometry all the time, to the extent that I often tell students that physics is just applied trigonometry. For any engineering discipline, classical physics is a must, and with it comes trigonometry.

If you separate sine and cosine from their uses with triangles, they are exciting functions in their own right.  Essentially, the sine and cosine functions describe sound waves, with the period defining the tone and the amplitude determining the loudness.  While you may not need to know this to play an instrument, you need to know something about this to build guitar effects pedals or do sound engineering.

The same types of waves are also used in communication.  The underlying signals that cell phones, radios, and other electronic equipment use to communicate wirelessly are based on the properties of sine and cosine.

However, even if your student doesn’t go into a technical field, the intellectual tools of trigonometry are essential.  Trigonometry helps your mind practice the art of choosing the correct intellectual tool to get a result.

While there are more rules than this, there are several essential tools that trig students use over and over again:

  • Knowing that the angles of a triangle add up to 180 degrees
  • Knowing that a line is also 180 degrees
  • Knowing that a right angle is 90 degrees
  • Knowing that a right triangle has one 90 degree angle
  • Being able to draw a line to create a right triangle
  • Then, with right triangles specifically:
    • The Pythagorean theorem
    • The standard ratios of sides (sine, cosine, tangent)
    • The inverses of the ratios (arcsine, arccosine, and arctangent)

Trigonometry is all about seeing the problem, identifying your tools, knowing what your tools do, and knowing how to link them together to get a result.  Trigonometry is more open-ended than other forms of mathematics.  Often, there is more than one way to get an answer to a trig problem.  The goal is to mentally craft a path from the problem to a solution using the tools in your toolbox.

So, I have no idea what path your student will go down.  However, I know that in almost any discipline, there will be problems to solve.  Every field has a “toolbox” of ideas, formulas, techniques, etc., which people use to solve problems.  Taking a problem, identifying an appropriate tool (or combination of tools) to use, and then getting a result is a skill that students can practice.  

Since we don’t know which discipline that will be, trigonometry gives us a great starting point for practicing the mental skillset using triangles as a training ground.  Your student’s future may involve triangles or sine waves.  But, even if it doesn’t, trigonometry will give them the mental practice they need to solve the problems they will face in the future.  

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