At least conceptually, I understand the relevance of algebra, calculus and statistics to our hi-tech, data-driven world.

But what is the relevance of trigonometric functions? Other than measuring the height of flagpoles by the shadows they cast, etc., what does an understanding of sine or tangents or cosecants really allow?

Seriously, most people don’t use trigonometry in their daily lives. Architects, carpenters, and some engineers will, but for the rest of us, it’s just a good exercise in deductive reasoning.

Trigonometric functions can describe periodic phenomena. With a bit of tweaking things as diverse as electromagnetic waves to seasonal temprature variation can be described using sine fuctions like Asin(Bx-C) where A represents amplitude, 2π/B is the period, and C is the phase shift.

Ultrafilter’s dead-on. As a structural engineer, I use trig waaay more often than any other course I took in school, including all the major specific courses. Having a basic understanding of the trig funtions really helps you “get” how things fit together.

What, Carnac-- you never need to calculate the height of a flagpole by measuring its shadow? Millions of dollars depend on it!

Try doing the opposite—figuring out how far the shadow will fall, with the flagpole at a certain height. I did that today.Except it wasnt a flagpole casting a shadow–it was a 30 story building standing (maybe) in the flight path of a small airport. The allowable height of the building is determined by the angle of the airplanes taking off .

Sine of the angle multiplied by the distance from the runway gives you the height of the building.
A multi-million dollar decision on real estate investment depends on my calculation.How’s that for making high-school math relevant?

Suppose you want to build a bookshelf in your room. You want to make it as tall as possible, but you realize that if your room has eight foot ceilings (say), and you build an eight-foot-high bookshelf, you won’t be able to stand it upright: it will hit the ceiling. What’s the largest bookshelf you can build and still be able to stand it upright?

This question actually came up when I was a kid and my dad was building a bookshelf for my sister. She was taking trig and my dad made her do the calculation to figure out how big to make the bookshelf. I thought, “Wow, that’s way cool! You can actually use math to figure stuff out!”

Much later I became a physicist and had to use math all the time for things like this.

There’s a dual purpose to learning things like trig in school. For the pupils who will one day be working in any of the fields already mentioned, it’s an essential basic grounding. For everyone else, it’s a useful way to improve one’s ability to manipulate numbers, and also (provided it’s taught well) is an excellent marker of problem-solving abilities.

What of the numerous other trigonometric funstions not listed. Can it be assumed that you understand them completely?
What of the natural numbers 'pi, and ‘e.’
what of the transcendental funcitons
What of LaPlacian Transforms, Bessel Functions, Complex Variables, etc. etc.
All of the above functions enter in to the solution of problems in all areas of business, manufacturing, distribution, etc. etc.

Just to toss out another everyday-life use that 99% of the world doesn’t know about, as a crime scene examiner in training we use sine, cosine, and tangent laws quite a bit to look at blood spatter stains on surfaces and calculate the possible 3D point of origin for each attack. We use this to reconstruct the events at the scene and to point to the truth or falsity of a witness’s, victim’s, or suspect’s police statement.

Using my extensive knowledge of trig, I have calculated that you need to build two 4-ft tall bookcases. Install the bottom one, then fit the top one in place. Voilà! No tilting needed!

Naah. Just find the building’s janitor and ask him how tall the flagpole is.

How 'bout a nice trig book instead? Everybody knows barometers are best used for finding the heights of buildings. First, you tie the barometer to a long string, lower it from the roof, then measure the amount of string used…

A very quick, back of the envelope calculation tells me that your shelves will be only about 3.5 inches deep. Just enough room for your paperbacks, I guess.