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I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trigonometry.

Are any purely theoretical uses for trigonometry? More specifically, can trigonometric concepts (or even functions) be used to prove/disprove general mathematical conjectures?

I have been told it is used a lot in calculus, but by my (extremely) limited knowledge it mainly consists of applying calculus concepts to trigonometric functions. Is this correct?

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    $\begingroup$ Trig functions are ubiquitous throughout all math calculus and above. Theory, applications, everything, very important to learn everything you can about trig functions if you plan to continue in math. $\endgroup$ – Merkh May 2 '16 at 0:13
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    $\begingroup$ "theoretical application" is kind of contradicting. $\endgroup$ – mvw May 2 '16 at 0:31
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    $\begingroup$ I meant applications to theoretical problems, such as proving theorems. $\endgroup$ – Conan G. May 2 '16 at 0:43
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    $\begingroup$ One theorem which can be proven using Fourier analysis (which is mentioned in an answer below) is the prime number theorem, which is important and non-trivial. The proof by Fourier analysis is not the standard proof, far as I know, but it astonishes me that sines and cosines could have anything to do with the distribution of prime numbers. $\endgroup$ – Milo Brandt May 2 '16 at 0:46
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    $\begingroup$ Another Fourier analysis proof is Hurwitz's proof of the isoperimetric inequality for rectifiable curves. So, from triangles comes a proof that circles enclose maximal area (of curves with given perimeter). $\endgroup$ – Eric Towers May 2 '16 at 12:31

14 Answers 14

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All the answers so far are concerned with applications in advanced mathematics, which is hardly comprehensible for high school students. Therefore, this is an answer that concentrates on applications on the high school level. It is going to be a long but hopefully exciting and thought-provoking journey.


Introduction

Although trigonometric functions seem to be concerned with only triangles, they do go far beyond that, even in elementary mathematics. Trigonometric functions are ubiquitous in mathematics. Due to lack of knowledge, however, trigonometric functions are usually applied in tricks or techniques when tackling problems that are otherwise difficult to solve. These methods exploit the properties of trigonometric functions and extend them to other concepts in mathematics. For instance, if you regard the sides of a triangle as merely positive real numbers, and by using some basic theorems, you can deduce statements of real numbers from their corresponding facts in trigonometry:

The law of cosines tells us that, in a triangle $ABC$, for any permutations of the sides $a, b, c$, $$c^2 = a^2 + b^2 - 2ab \cos C,$$ or equivalently, $$\cos C = \frac {a^2 + b^2 - c^2}{2ab}.$$

By using a well-known result from trigonometry, i.e. if $A, B, C$ are angles of a triangle, then $$\cos A + \cos B + \cos C \le \frac 32,$$ we have $$\frac {a^2 + b^2 - c^2}{2ab} + \frac {b^2 + c^2 - a^2}{2bc} + \frac {c^2 + a^2 - b^2}{2ca} \le \frac 32,$$ for $a, b, c$ sides of a triangle.

Remarks:

  1. The above inequality has various proofs (try to find one yourself!). You may find some of them here. However, the simplest of all these proofs is to use Jensen's inequality, which becomes easy once you know basic differential calculus and the concept of convexity.
  2. If the inequality seems trivial to you, we may go one step further, using something that is occasionally called Ravi's transformation. This trick is especially useful because it transforms sides of a triangle into pure numbers, discarding its geometrical significance. It states that a sufficient and necessary condition for $a, b, c$ to be the sides of a triangle is that there exist positive real numbers $p, q, r$ such that $$a = p + q,\, b = q + r,\, c = r + p.$$ (The proof is omitted here. Incidentally, the numbers $p, q, r$ here have geometrical interpretations. You can try to prove the theorem yourself, using geometry.) Now apply this to our inequality, and we get a statement about positive real numbers (instead of sides of a triangle). Moreover, after some algebraic manipulations the final inequality can seem as nontrivial as you like!

Trigonometric Substitution

The inverse of what we have done above, i.e. substituting algebraic expressions with trigonometric quantities, is a method called trigonometric substitution. In most cases, the problem does not even seem to be related to trigonometry at all; you will find it surprising how trigonometric methods lead to such beautiful proofs! Here is an example to illustrate this idea:

For $0 \lt a, b, c \lt 1$, if $ab + bc + ca = 1$, Prove that $$\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2.$$

Solution:

Observe that if $A, B, C$ are angles of a triangle, then $$\tan \frac A2 \tan \frac B2 + \tan \frac B2 \tan \frac C2 + \tan \frac C2 \tan \frac A2 = 1,$$

Let $a = \tan \frac A2, b = \tan \frac B2, c = \tan \frac C2$, where $A, B, C$ are angles of an acute triangle, taking into account the condition that $0 \lt a, b, c \lt 1$. Thus it suffices to prove that $$\frac {\tan \frac A2}{1 - \tan^2 \frac A2} + \frac {\tan \frac B2}{1 - \tan^2 \frac B2} + \frac {\tan \frac C2}{1 - \tan^2 \frac C2} \ge \frac {3 \sqrt 3}2,$$ or $$\tan A + \tan B + \tan C \ge 3\sqrt{3},$$ which is trivial. Q.E.D.

In this solution many results are used without proofs, but all of them are perfectly provable from trigonometric methods. Thus, we solved an algebraic statement using only trigonometric methods. Furthermore, if one were to prove this inequality from purely algebraic methods, then the solution will become tedious and ugly. If you are interested in this method, there is a good book for further reading. It contains a comprehensive chapter on these methods, as well as proofs of the identities and inequalities used here, of course.

Although we have restricted our attention to inequalities presently, the method of trigonometric substitution has much more applications. For example, it is an essential tool in the evaluation of integrals. You will encounter this technique in various situations when studying mathematics.


Parametrization

In the examples above we have exploited some properties of trigonometric functions, most of which are identities or inequalities involving the three angles of a triangle. Nonetheless, we need not restrict our focus to triangles only; we will look at the extended concept of trigonometric functions, i.e. for arbitrary real numbers.

An important property is that these functions have certain bounds. For example, $-1 \le \sin \theta \le 1$, for all $\theta\in\mathbb{R}$, and if $\theta \in [0, \frac {\pi}2)$ then $\tan \theta \gt 0$. We will use this property in the next example:

Consider two real numbers $x, y$ such that $$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1,$$ where $a, b$ are two given positive real numbers. Now, how do we find the maximum and minimum of, say, $x + y$?

Observe that the equation $x^2/a^2 + y^2/b^2 = 1$ just looks like $\sin^2 \theta + \cos^2 \theta = 1$! It is natural to substitute for $x, y$. Let $x^2/a^2 = \sin^2 \theta$ and $y^2/b^2 = \cos^2 \theta$ and we get the equations

\begin{cases} x = a \sin \theta, \\ y = b \cos \theta. \\ \end{cases}

(Note that $\theta$ can be any real number, though in this case you can restrict it to the interval $[0, 2 \pi]$.) So we need to find the extrema of $x + y$, that is, $a \sin \theta + b \cos \theta$, which is easy. The maximum is $\sqrt {a^2 + b^2}$ and the minimum $-\sqrt {a^2 + b^2}$.

Remarks:

  1. The attentive readers will find that this is essentially the same as trigonometric substitution. However, here this method is not considered as merely substitution. It is called parametrization, where $\theta$ plays the role of the parameter. The equations we obtained are called parametric equations.
  2. Surprisingly, not only curves like circles and ellipses but also lines have parametrizations involving trigonometric functions. A standard parametrization for a line that passes through the point $(a, b)$ with slope $\tan \theta$ will look like this: \begin{cases} x = a + t\cos \theta, \\ y = b + t\sin \theta. \\ \end{cases} The advantage of this parametrization is that $|t|$ equals the distance between the point $(x,y)$ and the given point $(a,b)$.

To conclude, we make some final remarks:

  1. Trigonometry is a bridge between geometry and (elementary) algebra: On the one hand, many theorems in geometry can be readily applied to algebraic problems; on the other hand, algebraic theorems can be used to prove geometrical statements.
  2. Trigonometric functions have many other applications, but most of them are too advanced for high school students. For example, we can use them to represent other periodic functions (a subject called Fourier series). Also, once you learn about Taylor series (when learning calculus) you will find that it even has a close connection with complex numbers, where you will see the beautiful formula (called Euler's formula) $$e^{ix}=\cos x+i\sin x.$$ This list may extend forever, as these functions are such fundamental in the edifice of mathematics.

Hope you had fun reading this!

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  • $\begingroup$ How to prove Euler's Formula? $\endgroup$ – Abhas Kumar Sinha Jul 27 '18 at 16:01
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I tell my students in differential equations (a one-semester-past-calculus class) that people have been lying to them about why trig is important! Solving triangles, who cares. Trig matters for all sorts of reasons in more advanced math.

One big deal is that the sine and the cosine both satisfy $$f(x+2\pi)=f(x);$$that is, they are functions with period $2\pi$ (in radians). The amazing and hugely important thing about that is that you can use sine and cosine to "generate" any other function with period $2\pi$. That is, leaving out a lot of technical details, if $f$ is any function with period $2\pi$ then there are constants $a_n$ and $b_n$ so that $$f(x)=a_0+(a_1\cos(x)+b_1\sin(x))+(a_2\cos(2x)+b_2\sin(2x))+\dots.$$ That's the "Fourier series" for $f$; Fourier series are awesomely useful and important in many areas of math, theoretical and applied both.


Confession: It appears people are reading this. I can't stand it; I told a lie above. For the record, one of the "technical details" I omitted is that it's not actually true that every $2\pi$-periodic function can be expanded in a Fourier series.

Fourier told the exact same lie. In some sense though it was true in his day, at least truer than it is now. Because the modern notion of "function" is very different from what people thought of as a "function" a few centuries ago. And it was one of the most fruitful lies in the history of mathematics.

In the same spirit, I think that although it's not actually true, in a pre-calculus context it's a more appropriate statement than any of the actually true assertions it approximates.

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    $\begingroup$ Couldn't you say "for any nice enough function" or "for any function you'll come across in real life" and not have to lie? Frankly, I'm satisfied with your original answer where "technical details" hides the unpleasantness that (IMO) we don't really need to worry about. $\endgroup$ – Teepeemm May 2 '16 at 19:01
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    $\begingroup$ @Teepeemm I certainly agree that in this context we don't need to worry about those details - that's why the original appeared containing the blatant lie in the first place. A little later I felt bad about spreading disinformation on the internet - having acknowledged the lie I'm not doing that. I kind of like the rhetorical whatever of the current version - notice I started by saying I tell my students they've been lied to, now it turns out I was lying somewhat as well, as did Fourier. I like the way the current version reads, and in this context I feel a good read is the main thing. $\endgroup$ – David C. Ullrich May 2 '16 at 22:14
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    $\begingroup$ @SimpleArt Because if you wrote it as $a_0+b_0 + \dots$, you could rewrite it as $c_0 + \dots$, where $c_0=a_0+b_0$, and then rename $c_0$ to $a_0$. Having two constants added together gives no more power. In fact, it's harmful, since it means that there are infinitely many ways to express $f$ (e.g., $1+2+\dots$, $0+3+\dots$, $\mathrm{e} + (3-\mathrm{e})+\dots$, ...). With a single constant, any suitable function $f$ has a unique expression as a sum of sines and cosines. $\endgroup$ – David Richerby May 3 '16 at 20:33
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    $\begingroup$ Speaking as a Physicist, if a function cannot be expressed as a Fourier series, it isn't worth it. $\endgroup$ – Davidmh May 4 '16 at 14:51
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    $\begingroup$ "Solving triangles, who cares". I'd interject that solving triangles is very important to classical mechanics. Also, I find that it is utterly astonishing how many things the ancient greeks were able to find out by pure abstraction and geometry - e.g., how Eratosthenes calculated the diameter of the Earth. $\endgroup$ – user159517 Dec 3 '17 at 11:51
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One momentous theoretical use of trigonometric functions is to solve the Basel problem (i.e. proving that the sum of the reciprocals of the squares equals $\pi^2/6$). See for example: http://math.cmu.edu/~bwsulliv/basel-problem.pdf.

Or better: https://en.wikipedia.org/wiki/Basel_problem#A_rigorous_elementary_proof (as suggested by Mark S. in comments).

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The functions sin, cos and $\exp(i \theta)$, and their applications, are everywhere in mathematics, but in a way that is relatively divorced from triangles.

Terms like "Fourier series/analysis/theory/transform", "trigonometric sums/series", "harmonic analysis" and "exponential sums" are used to indicate these not particularly geometric ways of using trigonometric functions. As an example, a lot of what is known about the distribution of prime numbers is based on this type of theory.

Trigonometry, the use of trigonometric functions in relation to triangles and geometry, appears where Euclidean geometry is in play, but that is not the case for most of higher mathematics. Applications such as engineering, (classical) physics, architecture, and surveying are based on Euclidean geometry in 2-3 dimensions, and there it can be necessary to use trigonometry of the kind taught in school. Spherical trigonometry also has its applications.

The end user of software that uses trigonometry may not ever have a need to calculate with trigonometric functions or solve triangles. It is mostly in physics and mechanical engineering that people have reason to constantly be using classical trigonometry.

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What I find amazing is the fact that $\cos x$ and identity involving it are tools used in a key step in a proof of the prime number theorem. I won't dive into technical details, but let me sketch the general idea of how it arises (taken from section 7.1.3 here)

The key part of every complex-analytic proof of PNT involves proving that $\zeta(1+iy)\neq 0$ for $y\in\Bbb R$. For that we consider auxilary function $h(x)=\zeta(x)^3\zeta(x+iy)^4\zeta(x+2iy)$. Using Euler's product and Taylor expansion of $\ln z$ around $1$ we can write $$\ln|h(x)|=\sum_{j=1}^\infty\sum_{n=1}^\infty\frac{p_j^{-nx}}{n}\operatorname{Re}(3+4p_j^{-iny}+p_j^{-2iny})$$

Now $p_j^{-iny}$ is a real number to a purely imaginary power, so its real part is $\cos\theta$, where $\theta=ny\ln p_j$. Similarly the real part of $p_j^{-2iny}$ is $\cos 2\theta$. Hence we have $$\operatorname{Re}(3+4p_j^{-iny}+p_j^{-2iny})=3+4\cos\theta+\cos 2\theta$$ and it's easy to check using $\cos 2\theta=2\cos^2\theta-1$ that RHS is nonnegative. Finally this tells us that $|h(x)|\geq 1$, which couldn't hold if $\zeta(1+iy)=0$.

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The "theory" aspect of trig functions gets a lot more interesting when you also bring in calculus, infinite series, and complex numbers. In fact at that point, you would usually define the sine and cosine functions as infinite series rather than by using geometry:

$$\begin{align}\sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\\ \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\end{align}$$

($3!$ means "$3$ factorial", i.e. $1\times2\times3$).

Those definitions work with $x$ in radians, not degrees ($2\pi$ radians = $360$ degrees), and for complex numbers, not just real values of $x$.

There are some deep relations between trig, exponential, and log functions, the most fundamental being $$e^{iz} = \cos z + i \sin z$$ which is true for any complex number $z$, and leads to the surprising result (discovered by Euler) linking four (or five?) of the most "fundamental constants" in mathematics: $$e^{i\pi} + 1 = 0.$$

Pretty much everything in applied maths uses either trig or exponential functions somewhere - they are not limited to calculating angles in geometry.

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Here's a purely 'theoretical application' that is quite different from the other replies here: Trigonometric identities are used to help show that 'angle trisection' - one of the three classic 'impossible problems' from antiquity - is in fact impossible.

The proof in broad outline is as follows: (a) Straightedge-and-compass constructions can only reach numbers (i.e lengths of line segments, vis-a-vis Cartesian Geometry) constructible from the rationals using only rational operations and square-root extractions; (b) No root of an irreducible cubic polynomial can be so constructed; (c) The cosine of 60 degrees is rational; d) Applying trigonometric identities to the cosine of 60 degrees yields a cubic polynomial function of the cosine of 20 degrees, so the latter satisfies a(n irreducible) cubic polynomial; (e) THEREFORE an angle of 60 degrees cannot be trisected.

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Here is approximately the simplest question I know, not involving trig functions, whose answer involves trig functions.

Question: How many subsets of the $n$-element set $\{ 1, 2, \dots n \}$ are there whose size is divisible by $5$?

In other words, the question is to compute the sum

$$a_n = \sum_{k=0}^{\lfloor \frac{n}{5} \rfloor} {n \choose 5k}$$

where ${n \choose k}$ is a binomial coefficient.

The answer, remarkably, is the following. Let $(r_k, \theta_k)$ be the polar coordinates of the point $\left( 1 + \cos \frac{2 \pi k}{5}, \sin \frac{2 \pi k}{5} \right)$ where $k = 0, 1, 2, 3, 4$. For example $r_0 = 2, \theta_0 = 0$, while $r_1$ turns out to be the golden ratio and $\theta_1 = \frac{\pi}{5}$. Then

$$a_n = \frac{1}{5} \left( \sum_{k=0}^4 r_k^n \cos n \theta_k \right).$$

In particular, the leading term is $\frac{2^n}{5}$ as you might guess, but the error in this approximation has some funny periodic behavior (of period $10$, rather than $5$) due to the presence of the cosine terms, with size described the golden ratio, which I think is quite hard to predict without knowing how to get this answer.

Understanding how this works will take you through a lot of fun and important mathematics, including but not limited to Euler's formula, de Moivre's formula, and the discrete Fourier transform. Ultimately the appearance of trig functions here is due to the appearance of complex numbers, which is a common way that trig functions appear in math, but by no means the only way.

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Allow me to be vague with some generality.

Exponentiation (and so logarithms) complex numbers and trigonometry are cohesively and beautifully connected by means of the Euler relation : $ e^{it} = \cos t + i \sin t $

The trio pervade all of mathematics whenever a change of direction is to be dealt with.

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Along with the many uses for trigonometric functions already given, they are useful in combinatorics, which is the art of counting how many things are in a set.

For example, it is a beautiful fact that $$\tan(x) + \sec(x) = 1 \cdot 1 + 1 \cdot x + 1 \cdot\frac{x^2}{2!} + 2\cdot \frac{x^3}{3!} + 5\cdot\frac{x^4}{4!} + 16\cdot\frac{x^5}{5!} + 61\cdot\frac{x^6}{6!} + \cdots$$

where $3! = 3 \cdot 2 \cdot 1$, $4! = 4 \cdot 3 \cdot 2 \cdot 1$, etc. The coefficients $$1, 1, 1, 2, 5, 16, 61, 272, \ldots$$ count the number of ways to arrange the numbers $1,2, \ldots, n$ in some order $c_1, c_2, \ldots$ so that $c_1 < c_2 > c_3 < c_4 > c_5$ and so on. So for example, the $5 \cdot \frac{x^4}{4!}$ comes from the five arrangements

$$1 < 3 > 2 < 4,$$ $$1 < 4 > 2 < 3,$$ $$2 < 3 > 1 < 4,$$ $$2 < 4 > 1 < 3,$$ $$3 < 4 > 1 < 2.$$

These are called alternating permutations.

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Yes, there are both real-world applications of trigonometry, and applications of trigonometry in solving purely theoretical problems in both mathematics and science.

First, there are basic applications such as:
1) Determining the height of a building using right triangle trigonometry
2) Finding the width of a river using the laws of sines and/or cosines

Then you get into deeper scientific applications such as:
3) Determining how the distance to star is using the skinny triangle
4) Modeling periodic phenomena with sine and cosine functions (electrical current)

Then there are applications that are important to our daily lives:
5) Communications technology which uses ttiangulation to determine the location of a signal
6) Sound signals are broken down into Fourier series of sine functions in order to manipulate each specific frequency for analysis or modification (music production)
7) Digital artists calculating angles for cameras or light rays or to construct 3D models used in movies for our entertainment.

Okay, maybe one would argue that (4) and (6) are not really trigonometry since you aren't specifically using triangles, but whatever...

Then there are more theoretical "applications":
8) The triangle inequality! (does that count?)
9) Trig functions pop up everywhere in higher math and science courses
10) Waves and other periodic phenomena
11) Many studies involving non-Euclidean geometry will consider triangles (the angles of which don't add up to $180^\circ$).

The list could go on forever. Triangles are everywhere.

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The expression $e^{i\theta} = \cos \theta + i \sin \theta$ has has been used repeatedly to prove theorems in number theory, complex analysis, probability, fourier analysis, astronomy, and more.

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As others have mentioned there is an equation discovered by Euler stating $ e^{(i*θ)}=cos(θ)+i*sin(θ)$

$ e^{(i*pi)}=-1 $ Is a beautiful result.

It also allows solutions to thing of the form $i^i$ due to $i=e^{i(pi/2+2k)}$ for natural numbers k.

One of my favorite results of this equation comes from the result that a transcendental number to a algebraic power is transcendental. Algebraic meaning is the root to some polynomial with Integer coefficients. Due to $ e^{(i*pi)}=-1 $ it is quickly verified that pi is transcendental. There are very few numbers of meaning to have a proof of transcendentalism even though they are conjectured. Examples of non proven ones are the Euler–Mascheroni constant and Apery's constant.

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To what may be your surprise, you can actually solve or factor polynomials using trig (identities).

For example:

$$0=x^3-3x+1$$

It can be factored into

$$0=(x-r_1)(x-r_2)(x-r_3)$$

where

$$r_k=2\cos\left(\frac{2\pi}9+\frac{2k\pi}3\right)\qquad k=1,2,3$$

Factoring using radicals can also be done for this polynomial can also be done, but $a)$ it won't have such a neat pattern and $b)$ in many cases it is far more complicated, for example, it will involve complex numbers.

It is true, in fact, that it is impossible to factor the above cubic without resorting to either lots of complex numbers, or resorting to the usage of trigonometric functions. If you compared the two side-by-side, it is almost without question which solution is the 'better' and 'more practical' solution, both for numerical purposes and simplicity.

Of course, if you don't believe me, you could try working out the algebra using lots of trig identities or graphing. $\ddot\smile$

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