What do sine, tan, cos actually mean?

Solution 1:

I think the historical reason for the confusion stems from graphing trigonometric functions in polar form versus rectangular form. In rectangular form, the following statement below is true. $$ \theta \quad =\quad x $$ (Where the meaning of this equality is that we let the measure of the angle on the unit circle be representative of the rectangular distance along the x-axis, or the domain, of the rectangular function.) But in polar form, this is not true, and $\theta$ means something different, namely, it is used to define the reference angle from zero radians, not the distance along the $x$-axis. Thus, the meaning of $\theta$ or $x$, depends on which way you are graphing in the two-dimensional plane, i.e., polar or rectangular plotting. In the polar method, the meaning of the rectangular coordinate pair $$(x,y)$$ is that it is an analogous location to the coordinate pair defined by the directed distance from the origin for a certain measure of $\theta$, with the abscissa, or $x$-value of the rectangular coordinate pair, defined by $$x=r\cdot \cos(\theta )$$ and the ordinate, or $y$-value of the rectangular coordinate pair, defined by $$y=r\cdot \sin(\theta )$$

In polar method, we therefore represent the coordinate pair, or point on the plane, using a different meaning of $\theta$, namely, as an input value for the angle from the origin, in terms of $r$ defined as a function of $\theta$, and thus we commonly refer to each coordinate pair on the plane with: $$(r,\theta )$$

If, instead, we intend to graph on the two-dimensional plane in rectangular form, then we let: $$ \theta \quad =\quad x $$ (Where $\theta$ is now being understood to be the rectangular distance along the $x$-axis, or the domain of the rectangular function.) Thus, instead of representing the angle measure from the zero radians, the angle theta now represents the distance along the $x$-axis from the origin. Additionally, in the case of the parent function derived from the unit circe, since: $$r=1$$ we can construct the following function machine of a single variable for rectangular graphing: $$ r\cdot \sin(\theta )=y $$ And this one is intuitive because the ordinate, or y variable, winds up conveniently isolated. And since we are very accustomed to y being a function of x, it makes perfect sense to write this as: $$[r\cdot \sin(\theta )=y=f(x)=\sin(x)]\quad \Longleftrightarrow \quad \theta =x\quad \cap \quad r=1$$ But where it gets less intuitively obvious is when we construct the function machine for rectangular graphing for cosine, which when we multiply both sides in your example by r: $$r\cdot \cos(\theta )=x$$ And this is strange because even though rectangular and polar graphing are referring to the same two-dimensional planes, this statement below is using x in two different ways. First, let's consider the logically equivalent statement: $$ [r\cdot \cos(\theta )=x=f(x)=\cos(x)]\quad \Longleftrightarrow \quad \theta =x\quad \cap \quad r=1 $$ The problem is that in the polar / rectangular conversion formula, we are defining $x$ as the abscissa value for the polar representation of a coordinate, but now that we are switching to rectangular graphing form, we are using $x$ as the input variable to represent theta. In short, we are using $x$ in two different ways. (Similarly, we are also using $y$ in two different ways.) Basically, we are replacing the polar usage of $x$ with the rectangular ordinate, or $y$, and we are replacing the polar usage of theta in the above statement with the rectangular usage of $x$ as an independent variable. This ambiguity can be removed if we replace our common way of writing the ratios in polar form, with dummy variables. This is why some teachers prefer using opposite, adjacent, etc., instead of $x$ and $y$ because we are using them in different ways when we construct the function machines. To remove the conflation, simply use phi for the polar abscissa formula, yielding: $$ [r\cdot \cos(\theta )=\varphi =f(x)= \cos(x)]\quad \Longleftrightarrow \quad \theta =x\quad \cap \quad r=1 $$ And thus both trigonometric ratios have been transformed into functions of $x$. Let's be clear, which $x$ do we mean!? Well, in this case, we mean $x$ as the independent variable and not the abscissa output of the polar representation's directed distance. The same can be mathematically deduced for all of the other functions. Again, it might be helpful to replace your original ratios with dummy variables that are defined as the abscissa and ordinate so as not to conflate the use of $x$ in the first case, which is used as an abscissa, with $x$ in the function machine, which, for cosine, has two different usages as described above. By so using dummy variables, the apparent multiple usages and meanings of $x$ are removed. This answers your first three questions. Understanding this properly, also sheds light on the fact that even in the construction of sine, and the other functions, we are still committing the same mathematical conceit of switching between uses of the variables $x$ and $y$ in the polar method that differ from uses of the variables $x$ and $y$ in the rectangular method. To be clear, these are uses, and we are still graphing in the two-dimensional coordinate plane in both cases.

Your next question is what is the relation of sine and $x$, and cosine and $x$, etc. There is a reason that trigonometric functions are called transcendentals. The relationship is transcendental. That means one cannot use basic arithmetic, i.e., addition, subtraction, multiplication, or division, to solve, for example: $$ \cos(30) $$ Now, in antiquity, this was painstakingly measured by iterations around the unit circle. As a result, tables were developed, notably in the case of Ptolemaic astronomy. Later, when the Calculus was developed, infinite series were used to approximate these transcendentals with more accuracy, and these are now collectively referred to as Taylor Series, which are MacLaurin Series that are not centered at 0. The relationship is therefore symbolic, and is that sine "of an angle" is measured by an ordinate on the unit triangle of a specific theta, or angle. Likewise, cosine "of an angle" is measured by the abscissa of the unit triangle of a specific theta, or angle. And so on for the others ...

You then additionally ask what the inverse function means and why we can write it. First off, in response to why we can write it, it should be noted that each trigonometric function machine requires a different domain and range restriction in order to be a function in the first place. This is why some camps of thought prefer to write "arc" before the function as opposed to using the negative one superscript which, to some, implies there is a perfect inverse function without restriction. Now, if you are asking more broadly why one can construct inverse functions, I would either defer to a theorist, or suggest more simply that sometimes one knows the ratio, but needs the angle, and sometimes one knows the angle, but needs the ratio. Consider early analysis of planetary movement, hot air balloon, calculating the distance of a kite string, determining the distance of a trajectory using parametric trig functions, and so on, of course all limited to two dimensions at present.

Then, the answer to your question about the meaning of the inverse function, is simply the logical converse, or arcsine "of a ratio (with $r=1$)" is measured by a specific theta on the unit triangle of a specific ordinate, or y-value. Likewise, arccosine "of a ratio (with $r=1$)" is measured by a specific theta on the unit triangle of a specific abscissa, or $x$-value. It is simply the logical converse function and meaning of the above trigonometric function machines (including the necessary restricted domains and ranges). Additionally, they are both transcendental and require either iterations or measurement, resulting in tables, or require approximations of transcendental values through the use of so-called Taylor Series, which are MacLaurin Series that are not centered at 0. This last part answers what the inverse function means.

Solution 2:

There is a $\theta$ on the right hand side. The definition of $\sin(\theta)$ is not just $y/r$; instead it is something like:

$y/r$, after you've drawn a right triangle with $\theta$ as an angle, and where $y$ is the length of the side opposite $\theta$ and $r$ is the length of the hypotenuse.

As you can see, that full definition does in fact contain a $\theta$, no less than $mx+c$ contains an $x$.

Solution 3:

Looking at your profile, you post a lot on StackOverflow, so maybe a programming analogy will help.

Say you want to make a Point class. There's a lot of ways you could do it. You could have member variables p.x and p.y, for the $x$ and $y$ coordinates (duh). That's the most common way. But in theory, you could also write it in terms of p.rad and p.theta. Why not? Or if you were feeling really perverse, you could use some kind of bipolar coordinate system. There's a lot of choices, and you don't want to have to re-write your code if you change your mind.

So, as programmers are wont to do, you introduce a layer of abstraction. All Point objects, however they're implemented, have the following methods: p.get_x(), p.get_y(), p.get_rad(), and p.get_theta(). So now you can safely forget about implementation details*. A Point doesn't necessarily have an x or a y anymore, no more so than it has a rad or a theta.

Now let's go back to math-land. Instead of p.get_x(), we have $x(p)$, and your formula becomes: $$ \sin \left( \theta(p) \right) = \frac{y(p)}{r(p)} $$

But since we're lazy, and the $p$s everywhere kinda clutter things up, we drop the $p$ from the notation. So whenever you see these kinds of statements, remember that there is an implicit "for any point $p$" in them. Hopefully this makes more sense now.

$\newcommand{\RR}{\mathbb{R}}$ EDIT: I guess I didn't really address what "sine" really means. It's just a function from $\RR$ to $\RR$ that's got certain properties. Nothing special about applying it to $\theta$.

*Other than performance concerns, of course. :)

Solution 4:

I guess it depends on how you define $\sin\theta$. One possible definition is $f(\theta)\equiv \theta- \frac{\theta^3}{3!}+\frac{\theta^5}{5!}...$ This series converges for all $\theta$ and is called $\sin\theta$. For more info look up Maclaurin series.

More generally I think you are confused about what a function is. The technical definition is daunting (at least for me) but what you put in can be different to what you get out. Consider a machine that assigns a number to the letters of the alphabet. $A\to 1$ here the domain of the function are letters whilst the range numbers.

The inverse function $\sin^{-1}$ will essentially assign the limit of the convergent series (or ratio) to an angle. The role of domain and range reversed. Note though that it is necessary to restrict the value of $\theta$ as multiple values converge to the same limit.

I hope this helps.