Transcription
Lecture Notes on PrecalculusEleftherios Gkioulekas
Copyright c 2010 Eleftherios Gkioulekas. All rights reserved.This document is the intellectual property of Dr. Eleftherios Gkioulekas and is made availableunder the Creative Commons License CC BY-SA /This is a human-readable summary of (and not a substitute for) the /4.0/legalcodeYou are free to: Share – copy and redistribute the material in any medium or format Adapt – remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.Under the following terms: Attribution – You must give appropriate credit, provide a link to the license, andindicate if changes were made. You may do so in any reasonable manner, but not inany way that suggests the licensor endorses you or your use. ShareAlike – If you remix, transform, or build upon the material, you must distributeyour contributions under the same license as the original.No additional restrictions – You may not apply legal terms or technological measures thatlegally restrict others from doing anything the license permits.Notices: You do not have to comply with the license for elements of the material in the publicdomain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessaryfor your intended use. For example, other rights such as publicity, privacy, or moralrights may limit how you use the material.These notes are constantly updated by the author. If you have not obtained this file from theauthor’s website, it may be out of date. This notice includes the date of latest update to thisfile. If you are using these notes for a course, I would be very pleased to hear from you, inorder to document for my University the impact of this work.The main online lecture notes website is: ou may contact the author at: drlf@hushmail.comLast updated: June 28, 2020
1Contents123456789Trigonometric identitiesPRE1: Review of geometryPRE2: Trigonometric functionsPRE3: Trigonometric identitiesPRE4: Trigonometric equations and inequalitiesPRE5: Application to TrianglesPRE6: VectorsPRE7: Sequences and seriesPRE8: Conic sections24134672104123143161
2Trigonometric identities
3Trigonometric identitiesa b sin(a b) sin a cos b sin b cos acos(a b) cos a cos b sin a sin btan a tan btan(a b) 1 tan a tan bcot a cot b 1cot(a b) (!!)cot b cot a 2a sin(2a) 2 sin a cos acos(2a) cos2 a sin2 a 2 cos2 a 1 1 2 sin2 a2 tan atan(2a) 1 tan2 acot2 a 1cot(2a) 2 cot a I sin(a b) sin(a b) sin2 a sin2 bI cos(a b) cos(a b) cos2 a sin2 b3a sin(3a) 4 sin3 a 3 sin acos(3a) 4 cos3 a 3 cos atan(3a) 3 tan a tan3 a1 3 tan2 aIn terms ofcos 2a 1 cos(2a)1 cos(2a)sin2 a cos2 a 221 cos(2a)1 cos(2a)22tan a cot a 1 cos(2a)1 cos(2a)tan(a/2) 2 tan(a/2)1 tan2 (a/2)sin a cosa 1 tan2 (a/2)1 tan2 (a/2)2 tan(a/2)1 tan2 (a/2)tan a cota 2 tan(a/2)1 tan2 (a/2)Transformation tosum 2 sin a cos b sin(a b) sin(a b) 2 cos a cos b cos(a b) cos(a b) 2 sin a sin b cos(a b) cos(a b)product a ba bsin a sin b 2 sincos22a ba bcos a cos b 2 coscos22a bb acos a cos b 2 sinsin(!!)22sin(a b)tan a tan b cos a cos bsin(b a)cot a cot b (!!)sin a sin bAlso note the factorizations:(π/2) a(π/2) acos22a (π/2 b)a (π/2 b)I sin a cos b sin a sin(π/2 b) 2 sincos22I 1 cos a 2 cos2 (a/2)I 1 cos a 2 sin2 (a/2)I 1 sin a sin(π/2) sin a 2 sin
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194ReferencesThe following references were consulted during the preparation of these lecture notes.(1) Pistofides (1988): “Algebra. I.”, unpublished lecture notes.(2) Pistofides (1989): “Algebra. II.”, unpublished lecture notes.(3) Xenou (1994): “Algebra and Analytic Geometry. 1”, , Ekdoseis ZHTH.(4) Xenou (1995): “Algebra B”, Ekdoseis ZHTH.Lecture notes by Pistofides are available for download athttp://www.math.utpa.edu/lf/OGS/pistofides.html
28.06.2020 · 194 References The following references were consulted during the preparation of these lecture notes. (1)Pisto des (1988): \Algebra. I.", unpublished lecture notes.
