Lecture Notes on Precalculus Eleftherios Gkioulekas
Transcript of Lecture Notes on Precalculus Eleftherios Gkioulekas
Lecture Notes on Precalculus
Eleftherios Gkioulekas
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The main online lecture notes website is: http://faculty.utpa.edu/gkioulekase/You may contact the author at: [email protected] 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 I.”, unpublished lecture notes.(2) Pistofides (1989): “Algebra. II.”, unpublished lecture notes.(3) Xenou (1994): “Algebra and Analytic Geometry. 1”, , Ekdoseis ZHTH.(4) Xenou (1995): “Algebra B”, Ekdoseis ZHTH.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