WEEK ONE
NUMBERS
Natural numbers:
- Positive whole numbers 1, 2, 3, …
Integers:
- Whole numbers -1, 0, 1, …
Rational numbers:
- Ratios of integers with non-zero denominator.
!
- Numbers of the form " where both n and m are integers, and m is non-zero integer.
Real numbers:
- All numbers on number line, including numbers like 𝜋 and √2.
COORDINATE GEOMETRY AND LINES
Distance formula:
$(𝑥# − 𝑥$ )# + (𝑦# − 𝑦$ )#
Gradient / slope:
%! & %"
(! & ("
Point-slope form of a line:
y - 𝑦$ = m(x - 𝑥$ )
y = mx + c
slop-intercept form of straight line:
Ax + Bx + C = 0, where A ≠ 0 and B ≠ 0
Two lines with gradients 𝑚$ and 𝑚# respectively, are parallel if 𝑚$ = 𝑚# , and are
&$
perpendicular if 𝑚$ 𝑚# = -1, or equivalently, 𝑚$ = " , provided 𝑚$ ≠ 0 and 𝑚# ≠ 0.
!
INEQUALITIES
Rules for inequalities:
1. Exactly one of the following is true: a < b, a = b, b < a
2. If a < b and b < c, then a < c
3. If a < b, then a + c < b + c
4. If a < b and c > 0, then ac < bc
5. If a < b and c < d, then a + c < b + d
6. If a < b and c < 0, then ac > bc
$ $
7. If 0 < a < b, then ) > *
For real numbers a, b, c and d:
1. If a ≤ b and b ≤ c, then a ≤ c
2. If a ≤ b, then a + c ≤ b + c
3. If a ≤ b and c ≤ d, then a + c ≤ b + d
4. If a ≤ b and c ≥ 0, then ac ≤ bc
5. If a ≤ b and c ≤ 0, then ac ≥ bc
$ $
6. If 0 < a ≤ b, then ) ≥ *
ABSOLUTE VALUE
𝑎 𝑖𝑓 𝑎 ≥ 0
|a| = 0
−𝑎 𝑖𝑓 𝑎 < 0
Which means that |a| is defined to be a when a ≥ 0 and is defined to be -a when a < 0.
,Properties of absolute values:
For all a, b 𝜖 R and n 𝜖 Z:
1. √𝑎# = |a|
2. |ab| = |a||b|
) |)|
3. | | = when b ≠ 0
* |*|
4. |an| = |a|n
5. If a > 0 then |x| = a if x = a or x = -a
6. |x| < a if -a < x < a
7. |x| > a if x > a or x < -a
8. |x| ≤ a if -a ≤ x ≤ a
9. |x| ≥ a if x ≥ a or x ≤ -a
10. |a + b| ≤ |a| + |b| this is the triangle identity
Proofs for triangle identity:
i) We have:
-|a| ≤ a ≤ |a|
-|b| ≤ b ≤ |b|
Hence, adding these two identities we get
-|a| + -|b| ≤ a + b ≤ |a| + |b|
⟺ - (|a| + |b|) ≤ a + b ≤ |a| + |b|
⟺ |a + b| ≤ |a| + |b|
ii) Since:
|a + b|2 = (a + b)2 = a2 + 2ab + b2
And
(|a| + |b|)2 = |a|2 + 2|a||b| + |b|2 = a2 + 2|ab| + b2
It follows that
(|a| + |b|)2 - |a + b|2 = 2|ab| - 2ab = 2(|ab| - ab)
And hence since |ab| ≥ ab we have that
(|a| + |b| - |a + b|)(|a| + |b| + |a + b|) = (|a| + |b|)2 - |a + b|2 ≥ 0
Therefore since (|a| + |b| + |a + b|) > 0; unless a = b = 0 (in which case the
identity is trivially true), it follows that
|a| + |b| - |a + b| ≥ 0
And hence that
|a + b| ≤ |a| + |b|
ANGLES
Use radians [rad] as unit for angles.
Relationship between radians and degrees is given by equation 180° = 𝜋rad.
,
It follows that an angle 𝜃 in degrees corresponds to 𝜃 rad in radians while an angle 𝜙 in
$-.
$-.°
radians corresponds to 𝜙 , 0)1 in degrees.
Note: when write an angle in radians we usually leave out the unit.
Conversion of some common angles.
TRIG FUNCTIONS
23345678 ;1<)=8!7
Sin 𝜃 = 9%3478!:58 Cot 𝜃 = 23345678
;1<)=8!7
Cos 𝜃 = 9%3478!:58
23345678
Tan 𝜃 = ;1<)=8!7
9%3478!:58
Sec 𝜃 =
;1<)=8!7
9%3478!:58
Csc 𝜃 =
23345678
,TRIG IDENTITIES
$
Csc 𝜃 = >?@ A
$
Sec 𝜃 = BC> A
$
Cot 𝜃 = DE@ A
>?@ A
Tan 𝜃 =
BC> A
BC> A
Cot 𝜃 = >?@ A
Sin2 𝜃 + cos2 𝜃 = 1
Tan2 𝜃 + 1 = sec2 𝜃
1 + cot2 𝜃 = csc2 𝜃
EVEN AND ODD IDENTITES
Sin(-𝜃) = -sin(𝜃)
Cos(-𝜃) = cos(𝜃)
PERIODIC IDENTITES
Since 𝜋 represents one full rotation around a circle we have:
Sin (𝜃 + 2𝜋) = sin(𝜃)
Cos(𝜃 + 2𝜋) = cos(𝜃)
ADDITION AND SUBTRACTION FORMULAS
DOUBLE-ANGLE FORMULAS
HALF-ANGLE FORMULAS
PRODUCT FORMULAS
, GRAPHS OF TRIG FUNCTIONS
sec(x)
NUMBERS
Natural numbers:
- Positive whole numbers 1, 2, 3, …
Integers:
- Whole numbers -1, 0, 1, …
Rational numbers:
- Ratios of integers with non-zero denominator.
!
- Numbers of the form " where both n and m are integers, and m is non-zero integer.
Real numbers:
- All numbers on number line, including numbers like 𝜋 and √2.
COORDINATE GEOMETRY AND LINES
Distance formula:
$(𝑥# − 𝑥$ )# + (𝑦# − 𝑦$ )#
Gradient / slope:
%! & %"
(! & ("
Point-slope form of a line:
y - 𝑦$ = m(x - 𝑥$ )
y = mx + c
slop-intercept form of straight line:
Ax + Bx + C = 0, where A ≠ 0 and B ≠ 0
Two lines with gradients 𝑚$ and 𝑚# respectively, are parallel if 𝑚$ = 𝑚# , and are
&$
perpendicular if 𝑚$ 𝑚# = -1, or equivalently, 𝑚$ = " , provided 𝑚$ ≠ 0 and 𝑚# ≠ 0.
!
INEQUALITIES
Rules for inequalities:
1. Exactly one of the following is true: a < b, a = b, b < a
2. If a < b and b < c, then a < c
3. If a < b, then a + c < b + c
4. If a < b and c > 0, then ac < bc
5. If a < b and c < d, then a + c < b + d
6. If a < b and c < 0, then ac > bc
$ $
7. If 0 < a < b, then ) > *
For real numbers a, b, c and d:
1. If a ≤ b and b ≤ c, then a ≤ c
2. If a ≤ b, then a + c ≤ b + c
3. If a ≤ b and c ≤ d, then a + c ≤ b + d
4. If a ≤ b and c ≥ 0, then ac ≤ bc
5. If a ≤ b and c ≤ 0, then ac ≥ bc
$ $
6. If 0 < a ≤ b, then ) ≥ *
ABSOLUTE VALUE
𝑎 𝑖𝑓 𝑎 ≥ 0
|a| = 0
−𝑎 𝑖𝑓 𝑎 < 0
Which means that |a| is defined to be a when a ≥ 0 and is defined to be -a when a < 0.
,Properties of absolute values:
For all a, b 𝜖 R and n 𝜖 Z:
1. √𝑎# = |a|
2. |ab| = |a||b|
) |)|
3. | | = when b ≠ 0
* |*|
4. |an| = |a|n
5. If a > 0 then |x| = a if x = a or x = -a
6. |x| < a if -a < x < a
7. |x| > a if x > a or x < -a
8. |x| ≤ a if -a ≤ x ≤ a
9. |x| ≥ a if x ≥ a or x ≤ -a
10. |a + b| ≤ |a| + |b| this is the triangle identity
Proofs for triangle identity:
i) We have:
-|a| ≤ a ≤ |a|
-|b| ≤ b ≤ |b|
Hence, adding these two identities we get
-|a| + -|b| ≤ a + b ≤ |a| + |b|
⟺ - (|a| + |b|) ≤ a + b ≤ |a| + |b|
⟺ |a + b| ≤ |a| + |b|
ii) Since:
|a + b|2 = (a + b)2 = a2 + 2ab + b2
And
(|a| + |b|)2 = |a|2 + 2|a||b| + |b|2 = a2 + 2|ab| + b2
It follows that
(|a| + |b|)2 - |a + b|2 = 2|ab| - 2ab = 2(|ab| - ab)
And hence since |ab| ≥ ab we have that
(|a| + |b| - |a + b|)(|a| + |b| + |a + b|) = (|a| + |b|)2 - |a + b|2 ≥ 0
Therefore since (|a| + |b| + |a + b|) > 0; unless a = b = 0 (in which case the
identity is trivially true), it follows that
|a| + |b| - |a + b| ≥ 0
And hence that
|a + b| ≤ |a| + |b|
ANGLES
Use radians [rad] as unit for angles.
Relationship between radians and degrees is given by equation 180° = 𝜋rad.
,
It follows that an angle 𝜃 in degrees corresponds to 𝜃 rad in radians while an angle 𝜙 in
$-.
$-.°
radians corresponds to 𝜙 , 0)1 in degrees.
Note: when write an angle in radians we usually leave out the unit.
Conversion of some common angles.
TRIG FUNCTIONS
23345678 ;1<)=8!7
Sin 𝜃 = 9%3478!:58 Cot 𝜃 = 23345678
;1<)=8!7
Cos 𝜃 = 9%3478!:58
23345678
Tan 𝜃 = ;1<)=8!7
9%3478!:58
Sec 𝜃 =
;1<)=8!7
9%3478!:58
Csc 𝜃 =
23345678
,TRIG IDENTITIES
$
Csc 𝜃 = >?@ A
$
Sec 𝜃 = BC> A
$
Cot 𝜃 = DE@ A
>?@ A
Tan 𝜃 =
BC> A
BC> A
Cot 𝜃 = >?@ A
Sin2 𝜃 + cos2 𝜃 = 1
Tan2 𝜃 + 1 = sec2 𝜃
1 + cot2 𝜃 = csc2 𝜃
EVEN AND ODD IDENTITES
Sin(-𝜃) = -sin(𝜃)
Cos(-𝜃) = cos(𝜃)
PERIODIC IDENTITES
Since 𝜋 represents one full rotation around a circle we have:
Sin (𝜃 + 2𝜋) = sin(𝜃)
Cos(𝜃 + 2𝜋) = cos(𝜃)
ADDITION AND SUBTRACTION FORMULAS
DOUBLE-ANGLE FORMULAS
HALF-ANGLE FORMULAS
PRODUCT FORMULAS
, GRAPHS OF TRIG FUNCTIONS
sec(x)
