Lecture Notes on Precalculus Eleftherios Gkioulekas
Transcript of Lecture Notes on Precalculus Eleftherios Gkioulekas
![Page 1: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/1.jpg)
Lecture Notes on Precalculus
Eleftherios Gkioulekas
![Page 2: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/2.jpg)
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 If you have not obtained this file fromthe author’s website, it may be out of date.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: [email protected] updated: February 1, 2015
![Page 3: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/3.jpg)
1
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
![Page 4: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/4.jpg)
2
![Page 5: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/5.jpg)
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)
3
![Page 6: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/6.jpg)
4
![Page 7: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/7.jpg)
5
![Page 8: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/8.jpg)
6
![Page 9: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/9.jpg)
7
![Page 10: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/10.jpg)
8
![Page 11: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/11.jpg)
9
![Page 12: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/12.jpg)
10
![Page 13: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/13.jpg)
11
![Page 14: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/14.jpg)
12
![Page 15: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/15.jpg)
13
![Page 16: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/16.jpg)
14
![Page 17: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/17.jpg)
15
![Page 18: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/18.jpg)
16
![Page 19: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/19.jpg)
17
![Page 20: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/20.jpg)
18
![Page 21: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/21.jpg)
19
![Page 22: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/22.jpg)
20
![Page 23: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/23.jpg)
21
![Page 24: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/24.jpg)
22
![Page 25: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/25.jpg)
23
![Page 26: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/26.jpg)
24
![Page 27: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/27.jpg)
25
![Page 28: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/28.jpg)
26
![Page 29: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/29.jpg)
27
![Page 30: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/30.jpg)
28
![Page 31: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/31.jpg)
29
![Page 32: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/32.jpg)
30
![Page 33: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/33.jpg)
31
![Page 34: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/34.jpg)
32
![Page 35: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/35.jpg)
33
![Page 36: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/36.jpg)
34
![Page 37: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/37.jpg)
35
![Page 38: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/38.jpg)
36
![Page 39: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/39.jpg)
37
![Page 40: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/40.jpg)
38
![Page 41: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/41.jpg)
39
![Page 42: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/42.jpg)
40
![Page 43: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/43.jpg)
41
![Page 44: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/44.jpg)
42
![Page 45: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/45.jpg)
43
![Page 46: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/46.jpg)
44
![Page 47: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/47.jpg)
45
![Page 48: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/48.jpg)
46
![Page 49: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/49.jpg)
47
![Page 50: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/50.jpg)
48
![Page 51: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/51.jpg)
49
![Page 52: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/52.jpg)
50
![Page 53: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/53.jpg)
51
![Page 54: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/54.jpg)
52
![Page 55: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/55.jpg)
53
![Page 56: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/56.jpg)
54
![Page 57: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/57.jpg)
55
![Page 58: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/58.jpg)
56
![Page 59: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/59.jpg)
57
![Page 60: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/60.jpg)
58
![Page 61: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/61.jpg)
59
![Page 62: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/62.jpg)
60
![Page 63: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/63.jpg)
61
![Page 64: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/64.jpg)
62
![Page 65: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/65.jpg)
63
![Page 66: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/66.jpg)
64
![Page 67: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/67.jpg)
65
![Page 68: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/68.jpg)
66
![Page 69: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/69.jpg)
67
![Page 70: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/70.jpg)
68
![Page 71: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/71.jpg)
69
![Page 72: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/72.jpg)
70
![Page 73: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/73.jpg)
71
![Page 74: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/74.jpg)
72
![Page 75: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/75.jpg)
73
![Page 76: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/76.jpg)
74
![Page 77: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/77.jpg)
75
![Page 78: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/78.jpg)
76
![Page 79: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/79.jpg)
77
![Page 80: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/80.jpg)
78
![Page 81: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/81.jpg)
79
![Page 82: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/82.jpg)
80
![Page 83: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/83.jpg)
81
![Page 84: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/84.jpg)
82
![Page 85: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/85.jpg)
83
![Page 86: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/86.jpg)
84
![Page 87: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/87.jpg)
85
![Page 88: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/88.jpg)
86
![Page 89: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/89.jpg)
87
![Page 90: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/90.jpg)
88
![Page 91: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/91.jpg)
89
![Page 92: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/92.jpg)
90
![Page 93: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/93.jpg)
91
![Page 94: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/94.jpg)
92
![Page 95: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/95.jpg)
93
![Page 96: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/96.jpg)
94
![Page 97: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/97.jpg)
95
![Page 98: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/98.jpg)
96
![Page 99: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/99.jpg)
97
![Page 100: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/100.jpg)
98
![Page 101: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/101.jpg)
99
![Page 102: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/102.jpg)
100
![Page 103: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/103.jpg)
101
![Page 104: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/104.jpg)
102
![Page 105: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/105.jpg)
103
![Page 106: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/106.jpg)
104
![Page 107: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/107.jpg)
105
![Page 108: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/108.jpg)
106
![Page 109: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/109.jpg)
107
![Page 110: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/110.jpg)
108
![Page 111: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/111.jpg)
109
![Page 112: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/112.jpg)
110
![Page 113: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/113.jpg)
111
![Page 114: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/114.jpg)
112
![Page 115: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/115.jpg)
113
![Page 116: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/116.jpg)
114
![Page 117: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/117.jpg)
115
![Page 118: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/118.jpg)
116
![Page 119: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/119.jpg)
117
![Page 120: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/120.jpg)
118
![Page 121: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/121.jpg)
119
![Page 122: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/122.jpg)
120
![Page 123: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/123.jpg)
121
![Page 124: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/124.jpg)
122
![Page 125: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/125.jpg)
123
![Page 126: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/126.jpg)
124
![Page 127: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/127.jpg)
125
![Page 128: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/128.jpg)
126
![Page 129: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/129.jpg)
127
![Page 130: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/130.jpg)
128
![Page 131: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/131.jpg)
129
![Page 132: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/132.jpg)
130
![Page 133: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/133.jpg)
131
![Page 134: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/134.jpg)
132
![Page 135: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/135.jpg)
133
![Page 136: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/136.jpg)
134
![Page 137: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/137.jpg)
135
![Page 138: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/138.jpg)
136
![Page 139: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/139.jpg)
137
![Page 140: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/140.jpg)
138
![Page 141: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/141.jpg)
139
![Page 142: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/142.jpg)
140
![Page 143: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/143.jpg)
141
![Page 144: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/144.jpg)
142
![Page 145: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/145.jpg)
143
![Page 146: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/146.jpg)
144
![Page 147: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/147.jpg)
145
![Page 148: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/148.jpg)
146
![Page 149: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/149.jpg)
147
![Page 150: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/150.jpg)
148
![Page 151: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/151.jpg)
149
![Page 152: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/152.jpg)
150
![Page 153: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/153.jpg)
151
![Page 154: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/154.jpg)
152
![Page 155: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/155.jpg)
153
![Page 156: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/156.jpg)
154
![Page 157: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/157.jpg)
155
![Page 158: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/158.jpg)
156
![Page 159: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/159.jpg)
157
![Page 160: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/160.jpg)
158
![Page 161: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/161.jpg)
159
![Page 162: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/162.jpg)
160
![Page 163: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/163.jpg)
161
![Page 164: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/164.jpg)
162
![Page 165: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/165.jpg)
163
![Page 166: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/166.jpg)
164
![Page 167: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/167.jpg)
165
![Page 168: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/168.jpg)
166
![Page 169: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/169.jpg)
167
![Page 170: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/170.jpg)
168
![Page 171: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/171.jpg)
169
![Page 172: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/172.jpg)
170
![Page 173: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/173.jpg)
171
![Page 174: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/174.jpg)
172
![Page 175: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/175.jpg)
173
![Page 176: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/176.jpg)
174
![Page 177: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/177.jpg)
175
![Page 178: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/178.jpg)
176
![Page 179: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/179.jpg)
177
![Page 180: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/180.jpg)
178
![Page 181: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/181.jpg)
179
![Page 182: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/182.jpg)
180
![Page 183: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/183.jpg)
181
![Page 184: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/184.jpg)
182
![Page 185: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/185.jpg)
183
![Page 186: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/186.jpg)
184
![Page 187: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/187.jpg)
185
![Page 188: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/188.jpg)
186
![Page 189: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/189.jpg)
187
![Page 190: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/190.jpg)
188
![Page 191: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/191.jpg)
189
![Page 192: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/192.jpg)
190
![Page 193: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/193.jpg)
191
![Page 194: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/194.jpg)
192
![Page 195: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/195.jpg)
193
![Page 196: Lecture Notes on Precalculus Eleftherios Gkioulekas](https://reader031.fdocuments.us/reader031/viewer/2022020203/5868c4a91a28abcd408bff5f/html5/thumbnails/196.jpg)
194
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