Lecture Notes on Precalculus - UTRGV

1 Lecture Notes on PrecalculusEleftherios GkioulekasCopyrightc 2010 Eleftherios Gkioulekas. All rights document is the intellectual property of Dr. Eleftherios Gkioulekas and is made availableunder the Creative Commons License CC BY-SA : is a human-readable summary of (and not a substitute for) the license: 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 licensor cannot revoke these freedoms as long as you follow the license 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 additional restrictions You may not apply legal terms or technological measures thatlegally restrict others from doing anything the license : 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.

2 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 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 If you have not obtained this file from theauthor s website, it may be out of date.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 main online Lecture Notes website is: may contact the author updated: June 28, 20201 Contents1 Trigonometric identities22 PRE1: Review of geometry43 PRE2: Trigonometric functions134 PRE3: Trigonometric identities465 PRE4: Trigonometric equations and inequalities726 PRE5: Application to Triangles1047 PRE6: Vectors1238 PRE7: Sequences and series1439 PRE8: Conic sections1612 Trigonometric identitiesTrigonometric identitiesa b2a sin(a b) = sinacosb sinbcosacos(a b) = cosacosb sinasinbtan(a b) =tana tanb1 tanatanbcot(a b) =cotacotb 1cotb cota(!)

3 !) = sin(2a) = 2 sinacosacos(2a) = cos2a sin2a= 2 cos2a 1 = 1 2 sin2atan(2a) =2 tana1 tan2acot(2a) =cot2a 12 cotaIsin(a+b) sin(a b) = sin2a sin2bIcos(a+b) cos(a b) = cos2a sin2b3a= sin(3a) = 4 sin3a+ 3 sinacos(3a) = +4 cos3a 3 cosatan(3a) =3 tana tan3a1 3 tan2aIn terms ofcos 2atan(a/2) sin2a=1 cos(2a)2cos2a=1 + cos(2a)2sina=2 tan(a/2)1 + tan2(a/2)cosa=1 tan2(a/2)1 + tan2(a/2)tan2a=1 cos(2a)1 + cos(2a)cot2a=1 + cos(2a)1 cos(2a)tana=2 tan(a/2)1 tan2(a/2)cota=1 tan2(a/2)2 tan(a/2)Transformation tosumproduct 2 sinacosb= sin(a b) + sin(a+b)2 cosacosb= cos(a b) + cos(a+b)2 sinasinb= cos(a b) cos(a+b) = sina sinb= 2 sina b2cosa b2cosa+ cosb= 2 cosa+b2cosa b2cosa cosb= 2 sina+b2sinb a2(!!)tana tanb=sin(a b)cosacosbcota cotb=sin(b a)sinasinb(!!)Also note the factorizations:I1 sina= sin( /2) sina= 2 sin( /2) a2cos( /2) a2 Isina cosb= sina sin( /2 b) = 2 sina ( /2 b)2cosa ( /2 b)2I1 + cosa= 2 cos2(a/2)I1 cosa= 2 sin2(a/2)34 PRE1: Review of geometry5678910111213 PRE2: Trigonometric functions1415161718192021222324252627282 93031323334353637383940414243444546 PRE3: Trigonometric identities474849505152535455565758596061 6263646566676869707172 PRE4: Trigonometric equations and inequalities7374757677787980818283848586 8788899091929394959697989910010110210310 4 PRE5: Application to Triangles1051061071081091101111121131141 15116117118119120121122123 PRE6: Vectors124125126127128129130131132133134 135136137138139140141142143 PRE7: Sequences and series1441451461471481491501511521531541 55156157158159160161 PRE8.

4 Conic sections16216316416516616716816917017117 2173174175176177178179180181182183184185 186187188189190191192193194 ReferencesThe 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 Notes by Pistofides are available for download