Lecture Notes on Precalculus

Eleftherios Gkioulekas

Copyright c©2010 Eleftherios Gkioulekas. All rights reserved.Permission is granted to make and distribute verbatim copies of this document, only on astrictly non-commercial basis, provided the copyright notice this permission notice, and theavailability information below are preserved on all copies.

These notes are constantly updated by the author. If you have not obtained this file fromthe author’s website, it may be out of date. This notice includes the date of latest update tothis file. 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: http://faculty.utpa.edu/gkioulekase/You may contact the author at: drlf@hushmail.comLast updated: February 1, 2015

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Contents

Trigonometric identities 1PRE1: Review of geometry 4PRE2: Trigonometric functions 13PRE3: Trigonometric identities 46PRE4: Trigonometric equations and inequalities 72PRE5: Application to Triangles 104PRE6: Vectors 123PRE7: Sequences and series 143PRE8: Conic sections 161

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Trigonometric identities

a± b 2a⇓ ⇓

sin(a± b) = sin a cos b± sin b cos acos(a± b) = cos a cos b∓ sin a sin b

tan(a± b) =tan a± tan b

1∓ tan a tan b

cot(a± b) =cot a cot b∓ 1

cot b± cot a(!!)

=⇒

sin(2a) = 2 sin a cos acos(2a) = cos2 a− sin2 a = 2 cos2 a− 1 = 1− 2 sin2 a

tan(2a) =2 tan a

1− tan2 a

cot(2a) =cot2 a− 1

2 cot a

I sin(a+ b) sin(a− b) = sin2 a− sin2 bI cos(a+ b) cos(a− b) = cos2 a− sin2 b

3a =⇒ sin(3a) = −4 sin3 a+ 3 sin acos(3a) = +4 cos3 a− 3 cos a

tan(3a) =3 tan a− tan3 a

1− 3 tan2 a

In terms of

cos 2a tan(a/2)

⇓ ⇓sin2 a =

1− cos(2a)

2cos2 a =

1 + cos(2a)

2sin a =

2 tan(a/2)

1 + tan2(a/2)cos a =

1− tan2(a/2)

1 + tan2(a/2)

tan2 a =1− cos(2a)

1 + cos(2a)cot2 a =

1 + cos(2a)

1− cos(2a)tan a =

2 tan(a/2)

1− tan2(a/2)cot a =

1− tan2(a/2)

2 tan(a/2)

Transformation to

sum product

⇓ ⇓

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)

=⇒

sin a± sin b = 2 sina± b

2cos

a∓ b

2

cos a+ cos b = 2 cosa+ b

2cos

a− b

2

cos a− cos b = 2 sina+ b

2sin

b− a

2(!!)

tan a± tan b =sin(a± b)

cos a cos b

cot a± cot b =sin(b∓ a)

sin a sin b(!!)

Also note the factorizations:

I 1± sin a = sin(π/2)± sin a = 2 sin(π/2)± a

2cos

(π/2)∓ a

2

I sin a± cos b = sin a± sin(π/2− b) = 2 sina± (π/2− b)

2cos

a∓ (π/2− b)

2I 1 + cos a = 2 cos2(a/2)I 1− cos a = 2 sin2(a/2)

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References

The 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 at

http://www.math.utpa.edu/lf/OGS/pistofides.html