Why do both sine and cosine exist?

Cosine is just a change in the argument of sine, and vice versa.

$$\sin(x+\pi/2)=\cos(x)$$
$$\cos(x-\pi/2)=\sin(x)$$

So why do we have both of them? Do they both exist simply for convenience in defining the other trig functions?

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You seem to be asking why we name them both, rather than why they exist, since the very relationships you’ve written shows that if one exists, the other does, too.

Essentially, all mathematical notation and names, except for a very small subset, are for convenience/clarity/human communication. Math does not require that we name any function consistently across separate proofs, but it becomes much easier to communicate and think about things when we do.

I prefer the relationship:

$$\sin(x)=\cos(\pi/2-x)\\\cos(x)=\sin(\pi/2-x)$$ since this is symmetric and obviously geometric when $0\leq x\leq \pi/2$, and because this is also the relationship between “tangent” and “cotangent” and “secant” and “cosecant.” It indicates a duality in these functions.

It is certainly something of a paradox that adding more names often simplifies our understanding. In particular, if you defined only one, it would give you a sense that one of these functions was “primary.” There is a hint of that error even in the names “sine” and “cosine,” which vaguely implies that “sine” is primary, but it would be particularly strong if we only defined “sine” and never defined “cosine.” We would have a harder time grasping the duality that happens in trig functions.

$$
\sin^2 \alpha + \cos^2 \alpha =1,
$$
$$
\sin^2 \alpha + \sin^2 \left( \alpha + \frac{\pi}{2} \right) =1.
$$
Which one do you prefer?

Post scriptum: of course this is not a deep answer, but I think that sometimes mathematicians prefer elegance to logical “economy”.

The question is ridiculous. (I’m afraid to say.)

It’s equivalent to asking “Why are there words for both North and South?”

(Of course you could just use “negative North” at all times, if desired.)

In my opinion, one would only ask the question at hand, if, one has rather naively “just noticed” – let’s put it that way – that sine and cosine are complementary.

Note too that, indeed, in English co-sine is simply “sine” … with the appropriate prefix!! Just as you’d expect.

Again to make analogy, one could ask questions such as “why do we label both matter and antimatter!” or “Why label both up and down?”

It’s clear, traditional, and expected in languages that there are matching terms for complementary qualities {rather than, let us say “minimalistically,” using only the one and then the negative of it} …

heaven and hell, paradis et enfer.

I think it’s conceptually cleaner to have both $\cos$ and $\sin$ as distinct notations, for the following reason: in my opinion, it’s a bit of a “coincidence” that each of the functions $\{\cos,\sin\}$ can be described as translations of the other.

My preferred definition of these two functions is the following: first, position yourself at $(1,0)$ on the unit circle. Then start walking anticlockwise at unit speed. It follows that if the time elapsed is $t$, your $x$-coordinate will be $\cos t$ and your $y$-coordinate will be $\sin t$.

But notice that, in general, this way of producing pairs of functions (namely: positioning yourself on a curve and walking at unit speed, and then projecting onto the $x$ and $y$ coordinate axes) won’t usually result in a pair of functions such that each can be defined as a translation of the other. The ability to define $\cos$ and $\sin$ in terms of each other is specifically a quirk of circles centered at the origin. In some sense, it’s kind of a coincidence.

By the way, this viewpoint explains why $\cos^2 t + \sin^2 t = 1$; it’s because you’re walking on the curve defined by $x^2+y^2= 1$. This also explains why $\cos(0) = 1$ and $\sin(0)= 0$; it’s because we positioned ourselves at $(1,0)$ to begin with. It also explains why $(\cos’ t)^2+(\sin’t)^2= 1$; it’s because you’re walking at unit speed. And finally, this explains why $\sin'(0)>0;$. It’s because we chose to start walking anticlockwise. I’m pretty sure these four conditions (listed below for your convenience) completely characterize the ordered pair ($\cos,\sin$) among all ordered pairs of differentiable functions $\mathbb{R} \rightarrow \mathbb{R}$.

  1. $\cos^2 t + \sin^2 t = 1$
  2. $\cos(0) = 1, \sin(0) = 0$
  3. $(\cos’t)^2+(\sin’t)^2 = 1$
  4. $\sin'(0) > 0$

Suppose you’re walking counterclockwise around the unit circle, starting at the point with coordinates $(1, 0)$. When you’ve walked a distance of $\theta$, your coordinates are $(\cos \theta, \sin \theta)$.

To me, that’s why it’s natural to name both cosine and sine: it takes two coordinates to describe a point in the plane, and both of those coordinates deserve names.

Mathematics is about making definitions that are as elegant and nice as possible, and allow your conclusions to easily or straightforwardly follow. In this case, it actually turns out the most elegant function, and the most helpful to understanding, is not cosine or sine. It is
$$
\textbf{cis}(x) = e^{ix} = \cos x + i \sin x.
$$
Of course, this is a complex-valued function–it takes in a real number $x$ and returns the point on the unit circle at angle $x$. But all of the properties you know about sine and cosine can be derived from this.

But more importantly, we see that $\cos$ and $\sin$ naturally arise in tandem, one as the real part and one as the imaginary part of $\text{cis}(x)$. So it makes little sense to define only one of the two without defining both.

Why do we define $\cos$ and $\sin$ at all, if we could just use $\text{cis}$? Because we often work with real numbers, and it is nice to have names for the real number functions arising from the complex exponential so that we don’t have to insert imaginary numbers into real-number expressions that eventually cancel out. Also for historical reasons.

I would suggest two reasons why we have both sin and cos.

First, historically, those who originally developed trigonometry started down the path of defining both functions, and then discovered various relationships between the two functions. Rather than redefine the work they did, we retain the functions as separate entities.

Second, for simplicity. The equation for a circular arc for example, can be more simply written as $x=\cos(t)$, $y=\sin(t)$, than as $x=\cos(t)$, $y=\cos(t-\pi/2)$.

If we are really trying to economize on functions, we could rewrite the tangent function $\tan(x)=\sin(x)/\cos(x)$ as $\cos(x-\pi/2)/\cos(x)$, but I think it would be more difficult to talk about the function this way, or to describe some of its remarkable properties, such as $\tan'(x)=\sec^2(x)$ or $\tan^2(x)+1=\sec^2(x)$.

From these two identities we can deduce: $\tan'(x) = \tan^2(x)+1$, which means $\tan(x)$ is a solution to the differential equation: $y’ = y^2+ 1$.

I’ll leave it to you to rewrite this derivation using $\cos(x)$ only.   🙂

Because $-1$ has no root in the reals.

Both sine and cosine can be defined as solutions to the differential equation
$$
f”(x) = -1 \cdot f(x).
$$
Which one you get depends only on the boundary condition.

Now, if the second derivative is so useful, then surely the first derivative should matter too! It’s tempting to write
$$
f'(x) = \sqrt{-1} \cdot f(x).
$$
and indeed it kind of works: it gives the complex function $\backslash t \mapsto e^{it}$. But for the applications the trigonometric functions were invented for – geometry, physics… – complex numbers can’t really be used, as in, you can’t measure a complex quantity.

To stay in the reals, we just leave the definition a second-order differential equation, but also give the first derivative a name of its own.

Before calculators and computers we had tables of values trig functions. Having all the trig functions: $\sin, \cos, \sec, \csc, \tan, \cot$ made doing geometric calculations easier. You didn’t have to first convert a cos into a sin before looking it up in the tables. It saved a step.

The $\sec$ and $\csc$ came into mathematical fashion when calculations for sailing ships across oceans were easier using them.

You can read more about the history of trig functions over thousands of years at http://www-history.mcs.st-and.ac.uk/HistTopics/Trigonometric_functions.html

They must both have names in order to preserve the symmetry when writing down relationships: in complex arithmetics ($e^{ix}=\cos x+i\sin x$) or in calculus ($\sin’ x=\cos x$), we see that they are two parts of the same coin, so if you only keep the name of one, expressions become lopsided and just plain ugly. You could just define one (and for the purpose of computing it numerically, you can certainly use just one to compute both), but in math, you want elegance.

Speaking of, a more pertinent question would be, why do $\csc$ and $sec$ exist. They have no real use except for creating confusion: they are fractions, don’t naturally arise from linear relationships (sin/cos are components of complex numbers, describe rotation, solutions to common differential equations) and, most importantly, they make otherwise simple expressions look unrecognizable and unreadable.

This is an elaboration on Vectornaut’s answer and hopefully more elementary than most others; actually two viewpoints. The explanations are kept elementary in order to emphasize the origin of calculations sin() and cos() and subsequent relations like the one in the question are based long after the original terms and are reflections of underlying symetries that people needed and wanted.

Consider any curve on a plane (or in space if you like the reasoning is the same. To embed it in a mathematical structure we can envision writing the curve paramatricized by s thusly (x(s),y(s). Now we use this technique to describe a circle with a parameterization of distance from the center and angle from x. We get:
$$\left(r\cdot \cos(\theta\right),r\cdot \sin\left(\theta\right))$$
In the abstract the terms are simply names for calculating the (x,y) coordinates for a particular curve. The identity mentioned is a reflection of the symmetry of that particular curve. If I had defined a hyperbola I would still have equations for (x,y) but they wouldn’t have that particular symmetry; and the question wouldn’t have been asked since the OP would see that two different functions would be expected with different relationships. All of the other properties of sin(),cos() are inherited from the underlying symmetry.
The alternate explanation comes from dropping the “r” dependence. Although this seems it would be more abstract but historically it was first 🙂
In ancient time people still had property boundaries and wanted to measure and design things remotely; say by drawing for communication. In order to layout property or build a temple some way of scaling had to be invented. Our ancestors found out the ratios of certain triangles were constant and therefore models or drawings could be scaled down; this can also be considered a symmetry. On areas of the earth that are relatively flat we have the ratios
$$ \left(\cos(\theta\right),\sin\left(\theta\right))$$
(They were as smart as we think we are)
Since they didn’t know any of the properties to start with they simply called the ratios different names. We discovered a multitude of properties later; over 1000’s of years.

I hope the readers will forgive the emphasis on symmetries but I am studying Lie Groups/Algebras and they seem to be advanced versions and descriptions of symmetries like the above.

Consider:

  • Why do we say up and down instead of just up, since down is just reversed up?
  • Why is liquid measured in liters when you could just use cubic decimeters?

Because each unit or word is appropriate to the thing it’s measuring.

A triangle has three sides. In a right triangle, each side’s length divided by another side’s length gives a ratio that reliably relates to one of the (non-right) angles of the triangle. You’ll notice that the permutations of 2 items taken 3 at a time is 6, which just happens to be how many basic trigonometric functions there are (not counting their inverses):

opposite / hypotenuse : sine        hypotenuse / opposite : cosecant
adjacent / hypotenuse : cosine      hypotenuse / adjacent : secant
opposite / adjacent   : tangent       adjacent / opposite : cotangent

Which of these ratios is not useful?

The fact that all of these can be determined from the others is irrelevant. By the same logic as your question, why use more than just one of the six names?

While they all describe the same thing, a triangle, they do it from useful, even if different, perspectives.

Having both $sin(x)$ and $cos(x)$ allows you to uniquely identify a point on the unit circle. $sin(x)$ is insufficient since you would also have to know $x$ to deduce $sin(\frac{\pi}{2} + x)$, which in turn renders $sin(x)$ useless since $x$ already uniquely identifies said point.

Just as every individual has a name , different from others ; so should every function have a name different from others .
This enables us to remember different functions in short, without having to write the detailed mathematical expressions , representting these functions.

It is similar to saying that why do spoon and fork exist even thought both of them do somewhat similar jobs….

Again yes there is a way to write cosine in terms of sine but if we write everything in sine the question which you will get will not look friendly. The more simple something is more is the motivation for you to do it.

Mathematics is all about making complicated things simpler not simpler things complicated