What Are Trig Functions, Really?

(An archive question of the week)

Trigonometric functions are sometimes introduced without a deep explanation of their meaning; they are just buttons to push on a calculator, or names to write in an equation. Even when a textbook gives a careful presentation, there are so many facets to the concept that it can be easy to get confused.

Functions, or ratios?

Here is a question from 2015 that gave me a chance to discuss all this:

Trig in the Age of Calculators, and Why Don't We Just Measure the Triangles Directly?

Hey! I'm having trouble understanding trig. I was fine with it till yesterday. Then we had a partnered assignment and my partner kind of confused me. 

I have a couple questions. 

1.) Are sine, cosine and tangent all functions that are applied to an angle which give the same answer as if you were to take the sides' ratios? 
 
2.) Why do we write SinA or CosA or TanA? This is more of a general question, I guess; but if we know the value of the angle, why don't we just solve the side with CosA or TanA or SinA? 

For example,

   Cos(82) = 0.94967769788

Why don't we solve Cos(82) as well? After all, we know that this side is the a/h ratio.

I understand what the inverse operations do and I understand the sine and cosine laws, but I'm having trouble getting my head around what sine and cosine and tangent *are*. We keep being told they're ratios, but I've read that they're functions ... so I'm assuming that the function is equal to the ratio? and if you apply the function to an angle, it doesn't matter if you know the sides or not, you can still get the sides' ratios? 

I'm really confused. I don't understand if to get the sine of something you do o/h, or if sine is a function and you input an angle and it gives you the same answer as if you were to take the opposite and hypotenuse of that angle and find their ratio. 

I think I need a slightly more detailed explanation of the trig ratios.

I’m not sure what question 2 means; I think it may be, “If we know the angle is 82°, don’t we already know the lengths of the sides and not need to calculate the cosine?”

So, what are the trig functions, really? Are they about angles, or triangles? Are they ratios, or functions? I first had to provide a definition (for which I  assumed Emily had learned only the right triangle definitions for acute angles):

I imagine a lot of students have questions like yours when they first learn trig, but they don't ask them! Let's get them cleared up.

Each trig function is defined as one of the ratios you would get if you made a right triangle using the given angle. (I'll talk a little later about how to improve this definition, but I'm starting where you presumably are.)

So given any acute angle, we are defining a FUNCTION based on a triangle we haven't actually made.

Is it really a function?

The first big question to ask is, are you sure this really defines a function? That is, for any input (angle) can we be sure that there is only one output value?

Let's focus on the sine function, to be specific. If we're given an angle, in order to find the sine of that angle by the definition, we need to make a right triangle. 

But which one? We could make infinitely many of them:

                 /
               +
             / :
           +   :
         / :   :
       /   :   :
     /     :   :
   +-------+---+---

Those are two of the possible triangles. The important thing is that the ratio of opposite to hypotenuse is the same for ALL of them, because they are all similar triangles. It doesn't matter how long we make the sides, provided we have the right angle. So given an angle, you'll always get the same value for its sine.

This is also explained here:

Sine, Cosine, Tangent

So we have a function that is defined as a ratio of sides of any right triangle with that angle; and we can prove that this is in fact a function of the angle, because it is independent of the size of the triangle we make.

How is it calculated?

But students often equate a function with a way of calculating it; for example, if we define a function as \(f(x) = 2x – 1\), that definition says we have to double x and decrease it by 1. The trig functions are not defined by such a mathematical process. They may, in fact, be the first functions they have seen that are defined so abstractly. (Really, though, the square root is defined similarly – it just seems more familiar.) So, how do we work with them?

The second question to ask is, how can we find the value of the function? We can't actually draw the triangle whenever we need to calculate sin(A), and that wouldn't be very accurate anyway. In the VERY old days (the ancient Greeks and maybe the Babylonians), they made tables of trig functions, probably by using some of the identities you will be learning, to derive the sine of a new angle from the sines they'd already calculated. 

You can find the sine of twice an angle using only the sine (and cosine) of the angle itself. And it turns out that the sine of a VERY small angle is very close to the radian measure of that angle (which, if you haven't learned, is the arc length of the angle on a unit circle: A*pi/180). There are also a couple special angles for which we know the exact value of the trig functions by geometry; we can start with those angles and work down to smaller angles.

Later table-makers used other methods that have been discovered over the years. For example, you can get a more accurate value for the sine of a small angle by using a few terms of this series (where x is the radian measure of the angle, and 3!, for example, means 3*2*1 = 6):

   sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

There are even quicker formulas, but they would be harder to describe.

In MY "old days," we looked up sines in such a table; today, our calculators do for us calculations similar to those done by the table-makers. But as far as we are concerned, we can imagine there is a genie in there drawing a really accurate triangle and measuring its sides for us. It gives the same result.

You may find this interesting:

  Methods of Computing Trig Functions
  http://mathforum.org/library/drmath/view/52576.html

So now we’ve defined some functions of an angle, and we’ve found ways to calculate them without drawing that triangle (in fact, far more accurately than we could ever measure); so we can start using them to do calculations about triangles in problems.

So the answer to most of your questions is "yes." The trig functions are functions (that are evaluated in complicated ways we don't need to know) that give the value of the appropriate ratio as if we had drawn the triangle. It is a function the value of which is defined by a ratio, but not usually evaluated using that ratio. Once you've evaluated the function, you can use it to find what the sides of a particular triangle would be much more accurately than if you measured them.

I'm not sure whether I've answered all your questions, especially the second; feel free to write back and ask more until we get it all clear.

Extending the definitions

Oh, I said I'd talk about a better definition. The definition in terms of right triangles applies only to acute angles, so we extend it further. One way to do that is to use a unit circle (radius 1) and say that, for example, the sine of any angle is the y-coordinate of the point where the terminal ray in standard position intersects the circle. For an acute angle, you can draw a triangle and see that this definition is identical to yours. But now we can use any angle at all. This makes trig much more useful. 

But you'll be getting there, if you haven't already.

This process is explained here:

Trig Functions, Reference Angles, and the Unit Circle

Addendum

A couple years later, another student asked a very similar question, which is not archived. Here are excerpts:

Ok. So whenever I ask this question to a math teacher, they say, "it's just a ratio." But they wouldn't go further into an explanation. So I decided to try and figure it out. So what is a sine and what is my calculator doing when I use sine in a problem, such as 8sin(34).

What confuses me the most is, what the heck is my calculator doing when I type in something like sin(25). Or just in general trigonometric functions.

At first, I thought a sine was just a number, like how the ratio of 3/4 is .75, and then you could use that number for the rest of the problem.

But then I realized that the ratio between (in this case for sines) the hypotenuse and the opposite side changes depending on what the angle is.

I gave the reference above, and summarized it briefly:

What is a sine? It is a FUNCTION of an angle, which is defined as a ratio of sides in a triangle with that angle. For any given angle, it gives you a number to use in whatever calculation you are doing, just as the square root gives you a number to use, that depends on the input.

What is your calculator doing? It is using one of several algorithms that give an accurate approximation of the ratio you would get if you made the appropriate triangle. (It can only be an approximation, because typically the result is irrational, and we can't write out the whole thing.)

He responded:

What is the function of the angle to the opposite side over the hypotenuse? Like how for squares y=x^2.
Let me rephrase that. How can I manipulate x so that when you plug in an angle degree for x, it will give you the ratio of opposite side over the hypotenuse?

This shows the common misunderstanding that a function must be a formula.

Perhaps you have missed a key point: a function does not have to have a formula. ANY relationship between two variables that yields a single value is a function, even if the only way to find that value is to look it up in a table or to measure something. I discussed some related issues here:

  Functions: The Very Idea
  http://mathforum.org/library/drmath/view/77548.html

I showed you, in the link I supplied last time, one way in which the value of the sine can be calculated, and a link for other methods. None of these ways is a "closed formula" in which you do a couple arithmetic operations and have an exact value; as I explained, they are all approximations that give the exact value only if continued forever. In a similar way, there is no closed formula for a square root, only algorithms that give values closer and closer to the exact value. But we push a button on a calculator and use that value. The same is true of trig functions. They are hard to do by hand, but the value you get is perfectly real.

I will probably be discussing that last reference sometime soon, along with several others about the nature of functions.