Transcription of Euler’s Formula and Trigonometry - Columbia University
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Euler’s Formula and Trigonometry - Columbia University
euler s Formula and TrigonometryPeter WoitDepartment of Mathematics, Columbia UniversitySeptember 10, 2019 These are some notes first prepared for my Fall 2015 Calculus II class, togive a quick explanation of how to think about Trigonometry using euler s for-mula. This is then applied to calculate certain integrals involving The sine and cosine as coordinates of the unitcircleThe subject of Trigonometry is often motivated by facts about triangles, but itis best understood in terms of another geometrical construction, the unit can defineDefinition(Cosine and sine).Given a point on the unit circle, at a counter-clockwise angle from the positivex-axis, cos is thex-coordinate of the point. sin is they-coordinate of the picture of the unit circle and these coordinates looks like this:1 Some trigonometric identities follow immediately from this definition, inparticular, since the unit circle is all the points in plane withxandycoordinatessatisfyingx2+y2= 1, we havecos2 + sin2 = 1 Other trignometric identities reflect a much less obvious property of thecosine and si
mulas. One can do this by showing that multiplication of a point z= x+ iy in the complex plane by ei rotates the point about the origin by a counter-clockwise angle . It then follows that multiplication by the product of ei 1 and ei 2 will be counterclockwise rotation by an angle 1 + 2, implying the correct exponential property ei 1ei 2 = ei( 1+ 2)
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