Euler’s Formula and Trigonometry

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

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Pour tout angle, on a la formule trigonométrique : 22 cos 60 sin 60 1 Donc : 2 sin 60 1 cos 60=1 122 1 1 4 1 3 2 4 4 4 4 §· ¨¸ ©¹ Soit : 3 sin60 0,866 4 De plus : sin60 0,866 tan60 1,732 cos60 0,5 IV - Démonstrations : Les formules de sinus, cosinus et tangente se démontrent avec le théorème de Thalès : Dans le triangle OFG : B OG

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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 numbersquotesdbs_dbs7.pdfusesText_5

[PDF] Euler’s Formula and Trigonometry - Columbia University

Euler’s Formula and Trigonometry - Columbia University