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Lecture Notes on Precalculus
Eleftherios Gkioulekas
c
Copyright 2010
Eleftherios Gkioulekas. All rights reserved.
Permission is granted to make and distribute verbatim copies of this document, only on a
strictly non-commercial basis, provided the copyright notice this permission notice, and the
availability information below are preserved on all copies.
These notes are constantly updated by the author. If you have not obtained this file from
the author’s website, it may be out of date. This notice includes the date of latest update to
this file. If you are using these notes for a course, I would be very pleased to hear from you, in
order 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: [email protected]
Last updated: February 1, 2015
1
Contents
Trigonometric identities
PRE1: Review of geometry
PRE2: Trigonometric functions
PRE3: Trigonometric identities
PRE4: Trigonometric equations and inequalities
PRE5: Application to Triangles
PRE6: Vectors
PRE7: Sequences and series
PRE8: Conic sections
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104
123
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2
3
Trigonometric identities
a±b
⇓
sin(a ± b) = sin a cos b ± sin b cos a
cos(a ± b) = cos a cos b ∓ sin a sin b
tan a ± tan b
tan(a ± b) =
1 ∓ tan a tan b
cot a cot b ∓ 1
cot(a ± b) =
(!!)
cot b ± cot a













2a
⇓
sin(2a) = 2 sin a cos a
cos(2a) = cos2 a − sin2 a = 2 cos2 a − 1 = 1 − 2 sin2 a
2 tan a
tan(2a) =
1 − tan2 a
cot2 a − 1
cot(2a) =
2 cot a
=⇒
I sin(a + b) sin(a − b) = sin2 a − sin2 b
I cos(a + b) cos(a − b) = cos2 a − sin2 b
3a =⇒
sin(3a) = −4 sin3 a + 3 sin a
cos(3a) = +4 cos3 a − 3 cos a
tan(3a) =
3 tan a − tan3 a
1 − 3 tan2 a
In terms of
cos 2a
⇓
1 + cos(2a)
1
−
cos(2a)
sin2 a =
cos2 a =
2
2
1 + cos(2a)
1 − cos(2a)
2
2
tan a =
cot a =
1 + cos(2a)
1 − cos(2a)
tan(a/2)
⇓
2 tan(a/2)
1 − tan2 (a/2)
sin a =
cos
a
=
1 + tan2 (a/2)
1 + tan2 (a/2)
2 tan(a/2)
1 − tan2 (a/2)
tan a =
cot
a
=
2 tan(a/2)
1 − tan2 (a/2)
Transformation to
sum
⇓

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±b
a∓b
sin a ± sin b = 2 sin
cos
2
2
a+b
a−b
cos a + cos b = 2 cos
cos
2
2
a+b
b−a
cos a − cos b = 2 sin
sin
(!!)
2
2
sin(a ± b)
tan a ± tan b =
cos a cos b
sin(b ∓ a)
cot a ± cot b =
(!!)
sin a sin b
Also note the factorizations:
(π/2) ± a
(π/2) ∓ a
cos
2
2
a ± (π/2 − b)
a ∓ (π/2 − b)
I sin a ± cos b = sin a ± sin(π/2 − b) = 2 sin
cos
2
2
I 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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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