[PDF] Euler’s Formula and Trigonometry

02 Force de Lorentz Force de Laplace

2e BC 2 Force de Lorentz Force de Laplace 11 Chapitre 2 : Force de Lorentz Force de Laplace 1 Expérience a) Dispositif expérimental Deux bobines de Helmholtz (2 bobines plates disposées parallèlement en regard, à la distance égale au rayon des bobines) créent un champ magnétique B uniforme parallèle à l'axe des bobines

La loi Normale ou loi de LAPLACE-GAUSS

Théorème de Laplace-de Moivre2, démontre qu'une telle fonction fournit la meilleure approximation possible d'une loi binomiale centrée réduite Les travaux datent de 1733 pour de Moivre et 1774 pour Laplace

7 Loi normale ou loi de Laplace-Gauss

Loi normale ou loi de Laplace-Gauss 45 7 « mathématiquement » prend des valeurs très faibles dès que l’on s’écarte suffisamment de μ: par exemple, une loi normale a seulement une chance sur un million de tomber au-delà de 5 écarts types de part et d’autre de la moyenne

Pouraugmenterlasurface A d’unfluidedansungazde dA

la loi de Laplace La bulle se dégonfle et minimise ainsi sa surface 7 L’accroissementdepression∆plorsquel’ontraverse une surface de séparation entre deux fluides dont les rayonsdecourburessontRetR0vaut P int −P ext = γ 1 R + 1 R0 7 Pourunesphère: P int −P ext = 2 γ R 7 Al’intérieurd’unebulledesavon P int = P 0 + 4 γ R 2

UE3-2 - Physiologie – Physiologie Respiratoire Chapitre 5

Loi de Laplace: P=2T/r, mais Tvarie en fonction de r Le surfactant abaisse plus la TS dans les petits alvéoles que dans les gros T A= T B/2 Propriétés élastiques

Chapitre V - Physique à Main Levée 300 expériences de

La loi de Laplace permet de calculer la différence pi –p0 = ∆p en fonction de R et de γ La surface d’une sphère vaut : S = 4πR2 Son augmentation dS est égale à : dS = 8πRdR Il s’ensuit : ∆pp p 2 i0R =− = γ La surpression ∆p est une fonction inverse du rayon de la goutte Si on augmente le rayon R de la goutte de dR, son

Moments, fonctions génératrices, trans- formées de Laplace

formées de Laplace 1 Moments et variance Théorème 6 1 Soit (;A;P) un espace de probabilité, et soit nun entier >0:Soit L nl’ensemble des v a Xsur cet espace telles que l’espérance m n= IE(Xn), appelée moment d’ordre n, existe Alors L nest un espace vectoriel, et on a L 1 ˙L 2 ˙˙L n:

Script Thermodynamique SPI (15h) 2005 06 d

¾ Notations et notions de calcul différentiel pour la thermodynamique ¾ Définition formelle de CP, CV, relation de Mayer pour le gaz parfait ¾ Définition de la fonction d’état Enthalpie H 7 Transformation réversible adiabatique ¾ Définition de γ ¾ Démonstration de la loi de Laplace

Euler’s Formula and Trigonometry

To understand the meaning of the left-hand side of Euler’s formula, it is best to recall that for real numbers x, one can instead write ex= exp(x) and think of this as a function of x, the exponential function, with name \exp" The true sign cance of Euler’s formula is as a claim that the de nition of the

[PDF] force de laplace exercices corrigés pdf

[PDF] force de lorentz exercice corrigé

[PDF] loi de laplace pdf

[PDF] force de laplace

[PDF] induction(correction exercice)

[PDF] propulsion fusée quantité de mouvement

[PDF] propulsion par réaction

[PDF] force de pression sur une paroi courbe

[PDF] force de pression sur une paroi plane tp

[PDF] force de pression sur une paroi inclinée

[PDF] force hydrostatique sur une surface courbe

[PDF] force de poussée hydrostatique

[PDF] force hydrostatique appliquée sur une paroi verticale plane

[PDF] quelle valeur ajoutée pensez vous pouvoir apporter

[PDF] décrivez votre personnalité exemple

Euler's Formula and Trigonometry

Peter Woit

Department of Mathematics, Columbia University

September 10, 2019

These are some notes rst prepared for my Fall 2015 Calculus II class, to give a quick explanation of how to think about trigonometry using Euler's for- mula. This is then applied to calculate certain integrals involving trigonometric functions.

1 The sine and cosine as coordinates of the unit

circle The subject of trigonometry is often motivated by facts about triangles, but it is best understood in terms of another geometrical construction, the unit circle.

One can dene

Denition(Cosine and sine).Given a point on the unit circle, at a counter- clockwise anglefrom the positivex-axis, cosis thex-coordinate of the point. sinis they-coordinate of the point. The picture of the unit circle and these coordinates looks like this: 1 Some trigonometric identities follow immediately from this denition, in particular, since the unit circle is all the points in plane withxandycoordinates satisfyingx2+y2= 1, we have cos

2+ sin2= 1

Other trignometric identities re

ect a much less obvious property of the cosine and sine functions, their behavior under addition of angles. This is given by the following two formulas, which are not at all obvious cos(1+2) =cos1cos2sin1sin2 sin(1+2) =sin1cos2+ cos1sin2(1) One goal of these notes is to explain a method of calculation which makes these identities obvious and easily understood, by relating them to properties of exponentials.

2 The complex plane

A complex numbercis given as a sum

c=a+ib wherea;bare real numbers,ais called the \real part" ofc,bis called the \imaginary part" ofc, andiis a symbol with the property thati2=1. For any complex numberc, one denes its \conjugate" by changing the sign of the imaginary partc=aib The length-squared of a complex number is given by cc= (a+ib)(aib) =a2+b2 2 which is a real number. Some of the basic tricks for manipulating complex numbers are the following: To extract the real and imaginary parts of a given complex number one can compute

Re(c) =12

(c+c)

Im(c) =12i(cc)(2)

To divide by a complex numberc, one can instead multiply byc cc in which form the only division is by a real number, the length-squared of c. Instead of parametrizing points on the plane by pairs (x;y) of real numbers, one can use a single complex number z=x+iy in which case one often refers to the plane parametrized in this way as the \com- plex plane". Points on the unit circle are now given by the complex numbers cos+isin These go around the circle once starting at= 0 and ending up back at the same point when= 2. Now the picture is3 A remarkable property of complex numbers is that, since multiplying two of them gives a third, they provide something new and not at all obvious: a consistent way of multiplying points on the plane. We will see in the next section that multiplication by a point on the unit circle of anglewill have an interesting geometric interpretation, as counter-clockwise rotation by an angle

3 Euler's formula

The central mathematical fact that we are interested in here is generally called \Euler's formula", and writtene i= cos+isinUsing equations 2 the real and imaginary parts of this formula are cos=12 (ei+ei) sin=12i(eiei)(which, if you are familiar with hyperbolic functions, explains the name of the hyperbolic cosine and sine). In the next section we will see that this is a very useful identity (and those of a practical bent may want to skip ahead to this), but rst we should address the question of what exactly the left-hand side means. The notation used implies that it is \the numbereraised to the poweri" and a striking example of this is the special case of=, which says e i=1 which relates three fundamental constants of mathematics (e;i;) although these seem to have nothing to do with each other. The problem though is that the idea of multiplying something by itself an imaginary number of times does not seem to make any sense. To understand the meaning of the left-hand side of Euler's formula, it is best to recall that for real numbersx, one can instead write e x= exp(x) and think of this as a function ofx, the exponential function, with name \exp". The true signcance of Euler's formula is as a claim that the denition of the exponential function can be extended from the real to the complex numbers, preserving the usual properties of the exponential. For any complex number c=a+ibone can apply the exponential function to get exp(a+ib) = exp(a)exp(ib) = exp(a)(cosb+isinb) 4 The trigonmetric addition formulas (equation 1) are equivalent to the usual property of the exponential, now extended to any complex numbersc1=a1+ib1 andc2=a2+ib2, giving e c1+c2=ea1+a2ei(b1+b2) =ea1+a2(cos(b1+b2) +isin(b1+b2)) =ea1+a2((cosb1cosb2sinb1sinb2) +i(sinb1cosb2+ cosb1sinb2)) =ea1(cosb1+isinb1)ea2(cosb2+isinb2) =ec1ec2 It is possible to show thatei= cos+isinhas the correct exponential property purely geometrically, without invoking the trigonometric addition for- mulas. One can do this by showing that multiplication of a pointz=x+iy in the complex plane byeirotates the point about the origin by a counter- clockwise angle. It then follows that multiplication by the product ofei1and e i2will be counterclockwise rotation by an angle1+2, implying the correct exponential property e i1ei2=ei(1+2) To show that multiplication byeiwill give a rotation by, one can argue as follows. One can easily see that multiplication byeirotates the pointz= 1 along the unit circle by an angle, taking (in terms of real coordinates) (1;0)!(cos;sin) This is also true for the pointz=i, which gets taken toi(cos+isin) = sin+icos. In terms of real coordinates on the plane, this is (0;1)!(sin;cos) and the rotation looks like this:5 An arbitrary point on the plane is a linear combination of the points (1;0) and (0;1), and one can see that multiplication byeiwill act as rotation by on any such linear combination, knowing that it does so for the cases of (1;0) and (0;1). Two other ways to motivate an extension of the exponential function to complex numbers, and to show that Euler's formula will be satised for such an extension are given in the next two sections.

3.1eias a solution of a dierential equation

The exponential functionsf(x) = exp(cx) forca real number has the property ddx f=cf One can ask what function ofxsatises this equation forc=i. Using the derivatives of the cosine and sine one nds ddx (cosx+isinx) =sinx+icosx=i(cosx+isinx) so cosx+isinxhas the correct derivative to be the desired extension of the exponential function to the casec=i.

3.2eiand power series expansions

By the end of this course, we will see that the exponential function can be represented as a \power series", i.e. a polynomial with an innite number of terms, given by exp(x) = 1 +x+x22! +x33! +x44! There are similar power series expansions for the sine and cosine, given by cos= 122! +44!
and sin=33! +55!
Euler's formula then comes about by extending the power series for the expo- nential function to the case ofx=ito get exp(i) = 1 +i22! i33! +44!
and seeing that this is identical to the power series for cos+isin. 6

4 Applications of Euler's formula

4.1 Trigonometric identities

Euler's formula allows one to derive the non-trivial trigonometric identities quite simply from the properties of the exponential. For example, the addition for- mulas can be found as follows: cos(1+2) =Re(ei(1+2)) =Re(ei1ei2) =Re((cos1+isin1)(cos2+isin2)) =cos1cos2sin1sin2 and sin(1+2) =Im(ei(1+2)) =Im(ei1ei2) =Im((cos1+isin1)(cos2+isin2)) =cos1sin2+ sin1cos2 Multiple angle formulas for the cosine and sine can be found by taking real and imaginary parts of the following identity (which is known as de Moivre's formula): cos(n) +isin(n) =ein =(ei)n =(cos+isin)n For example, takingn= 2 we get the double angle formulas cos(2) =Re((cos+isin)2) =Re((cos+isin)(cos+isin)) =cos 2sin2 and sin(2) =Im((cos+isin)2) =Im((cos+isin)(cos+isin)) =2sincos 7

4.2 Derivatives of trigonometric functions

Writing the cosine and sine as the real and imaginary parts ofei, one can easily compute their derivatives from the derivative of the exponential. One has dd cos=dd

Re(ei)

dd (12 (ei+ei)) i2 (eiei) =sin and dd sin=dd

Im(ei)

dd (12i(eiei)) 12 (ei+ei) =cos

4.3 Integrals of exponential and trigonometric functions

Three dierent types of integrals involving trigonmetric functions that can be straightforwardly evaluated using Euler's formula and the properties of expo- nentials are:

Integrals of the form

Z e axcos(bx)dxorZ e axsin(bx)dx are typically done in calculus textbooks using a trick involving two inte- grations by parts. They can be more straightforwardly evaluated by using Euler's formula to rewrite them as integrals of complex exponentials, for 8 instance Z e axcos(bx)dx=Re(Z e axeibxdx) =Re( Z e (a+ib)xdx) =Re(

1a+ibe(a+ib)x) +C

=Re( aiba

2+b2eaxeibx) +C

=Re( aiba

2+b2eax(cos(bx) +isin(bx))) +C

1a

2+b2eax(acos(bx) +bsin(bx)) +C

Integrals of the form

Z cos(ax)cos(bx)dx;Z cos(ax)sin(bx)dxorZ sin(ax)sin(bx)dx are usually done by using the addition formulas for the cosine and sine functions. They could equally well be be done using exponentials, for instance (assuminga6=b) Z cos(ax)cos(bx)dx=Z12 (eiax+eiax)12 (eibx+eibx)dx 14 Z (ei(a+b)x+ei(ab)x+ei(ab)x+ei(a+b)x)dx 12 Z (cos((a+b)x) + cos((ab)x))dx 12 (1a+bsin((a+b)x) +1absin((ab)x)) +C Whenaandbare integersm;n, and one integrates over an interval of size 2(for instance [;]), the above integrals give very simple results.

This is due to the fact that

Z eimxeinxdx=Z ei(m+n)xdx=(

0 ifm6=n

2ifm=n

One can show this by integrating the exponential, or more simply by noticing that the real and imaginary parts of the answer will, form6=n, be given by integrating a cosine and sine overm+nperiods, This gives zero since the area under the curves is the same above and below thex- axis. Form=n, the integrand is just 1, so the integral is the length of the interval of integration. 9

Integrals of the form

Z cos mx dx;Z cos mxsinnx dxorZ sin mx dx are performed in calculus textbooks by a combination of use of the sub- stitutionu= sin(x) oru= cos(x), of the identity cos2x+ sin2x= 1 to turn even powers of the cosine into even powers of the sine (and vice- versa), as well as the double angle formulas for the cosine and sine. Such methods are often the simplest ones, but one can also do such integrals by expressing them in terms of exponentials. For example Z cos

3x dx=Z

(12 (eix+eix))3dx 18 Z (e3ix+ 3eix+ 3eix+ei3x)dx 14 Z (cos(3x) + 3cos(x))dx 112
sin(3x) +34 sin(x) +C Note that this technique will typically give answers in a dierent form than the technique used in the book, giving not powers of the cosine or the sine, but something equivalent related to these by multiple-angle formulas.

4.4 Polar coordinates

Instead of Cartesian coordinatesxandy, one can parametrize points in the plane by polar coordinatesr(the distance from the origin) and(the angle with the positivexaxis. A point with polar coordinates (r;) has (x;y) coordinates x=rcos; y=rsin10 The point with polar coordinates (r;) has a simple complex coordinate, which, using Euler's formula, will be given by z=x+iy=rcos+irsin=rei

5 Problems

1. Compute

Z e axsin(bx)dx for arbitrary real constantsaandb.

2. Compute

Z cos(ax)sin(bx)dx for arbitrary real constantsaandb.

3. Compute

Rcos3x dxusing Euler's formula, and show that the result is the same as what one gets by the textbook method (using a substitution u= sinx). 11quotesdbs_dbs6.pdfusesText_12