Things You Should Know (But Likely Forgot), page 1 of (
Things You Should Know (But Likely Forgot)
Trigonometric identities:
cos# )  sin# ) œ "
tan# ) œ sec# )  "
cosa# ) b œ cos# )  sin# ) œ # cos# )  " œ "  # sin# )
sina#) b œ # cos ) sin ) ß
cos# ) œ "# a"  cosa# ) bbß
sin# ) œ "# a"  cosa# )bb
cosa+ „ ,b œ cosa+bcosa,b … sina+bsina,b
sina+ „ ,b œ sina+bcosa,b „ cosa+bsina,b
sina+bcosa,b œ "# asina+  ,b  sina+  ,bb
cosa+bsina,b œ "# asina+  ,b  sina+  ,bb
cosa+bcosa,b œ "# acosa+  ,b  cosa+  ,bb
sina+bsina,b œ "# acosa+  ,b  cosa+  ,bb
‰ ˆ +, ‰
sina+b  sina,b œ # sinˆ +,
# cos #
‰ ˆ +, ‰
sina+b  sina,b œ # cosˆ +,
# sin #
‰ ˆ +, ‰
cosa+b  cosa,b œ # cosˆ +,
# cos #
‰ ˆ +, ‰
cosa+b  cosa,b œ  # sinˆ +,
# sin #
3)
cos ) œ / /
#
3)
acos ) bw œ  sin )
atan ) bw œ sec# )
3)
sin ) œ / /
#3
3)
/„3) œ cos ) „ 3 sin )
asin )bw œ cos )
asec )bw œ sec ) tan )
Weierstrass substitution:
#
> œ tanˆ B# ‰ makes .B œ ">
# .>ß
#
">
cos B œ ">
#ß
Hyperbolics:
cos B even,
œ sin B odd,
/„B œ cosh B „ sinh B
B
B
cosh B œ / /
ß
#
cosh# B  sinh# B œ " ß
B
sinh B œ / /
#
B
ß
#>
sin B œ ">
#
cosh B even
sinh B odd
sinh B
tanh B œ cosh
Bß
tanh# B œ "  sech# B
"
sech B œ cosh
B
Things You Should Know (But Likely Forgot), page 2 of (
"Arctrig" (inverse trigonometric functions):
sin" aBb has domain c  "ß "dß
range   1# ß 1# ‘
tan" aBb has domain a  _ß _bß
range ˆ  1# ß 1# ‰
"B
"
acos aBbb œ È #
"B
w
"
"
atan aBbb œ "B
#
sinh" aBb œ lnŠB  ÈB#  "‹ß domain and range a  _ß _bß
asinh" aBbb œ ÈB"# "
cos aBb has domain c  "ß "dß
"
asin" aBbb œ È " #
w
range c !ß 1d
"
w
Inverse hyperbolics:
cosh" aBb œ lnŠB  ÈB#  "‹ß domain Ò"ß _Ñß range Ò!ß _Ñ
tanh" aBb œ "# ln "B
"B ß domain a  "ß "bß range a  _ß _b
w
acosh" aBbb œ È "#
w
"
atanh" aBbb œ "B
#
w
Useful integration facts and tricks:
B "
' 0 ÐBÑ .B œ '0B 0 Ð>Ñ .>  G for a 0 of your choice (at which the expression is
'
? .@ œ a?@bk,+ 
+
,
'+ @ .? ÞÞÞintegration by parts
well-defined)
,
' B/B .B œ aB  "b/B  G
"
' +,B
.B œ  +" ln k+  ,Bk  G
"
' + B
.B œ +" tan" ˆ B+ ‰  G
'
#
#
"
"
+B ¸
.B œ #+
ln¸ +B
G
+# B#
'È"
+# B#
.B œ sinh" ˆ B+ ‰  G
' ln B .B œ BalnB  "b  G
'È"
+# B#
.B œ sin" ˆ B+ ‰  G
'È"
B# +#
.B œ cosh" ˆ B+ ‰  G
One variable at a time:
`
if 0 aBß C b œ `C
J aBß C b then ' 0 aBß C b.C œ J aBß C b  1aBb
where 1 is an arbitrary function of B.
It works with more variables: '
`J aBßCßD b
`C
.C œ J aBß Cß D b  1aBß D bÞ
Things You Should Know (But Likely Forgot), page 3 of (
Determinant tricks: for 8 ‚ 8 matrices,
Ô row" ×
Ô row" ×
Ö ã Ù
Ö ã Ù
Ö
Ù
Ö
Ù
detÖ > row4 Ù œ > det Ö row4 Ù
Ö
Ù
Ö
Ù
ã
ã
Õ row8 Ø
Õ row8 Ø
(multiplying a row by > multiplies the determinant by >)
Ô row" ×
Ô row" ×
Ö ã Ù
Ö ã Ù
Ö
Ù
Ö
Ù
Ö row3 Ù
Ö row4 Ù
Ö
Ù
Ö
Ù
det Ö ã Ù œ a  "bdetÖ ã Ù
Ö
Ù
Ö
Ù
Ö row4 Ù
Ö row3 Ù
Ö
Ù
Ö
Ù
ã
ã
Õ row8 Ø
Õ row8 Ø
(swapping any two rows flips the sign of the determimamt)
row"
Ô
×
Ô row" ×
Ö
Ù
Ö ã Ù
ã
Ö
Ù
Ö
Ù
row3
Ö
Ù
Ö row3 Ù
Ö
Ù
Ö
Ù
ã
det Ö
Ù œ det Ö ã Ù
Ö
Ù
Ö
Ù
Ö row4  > row3 Ù
Ö row4 Ù
Ö
Ù
Ö
Ù
ã
ã
Õ
Ø
Õ row8 Ø
row8
(adding a multiple of one row to another row does not change the determinant)
Shortcut:
‡
‡ ×
Ô+ ‡
- â×
Ô ,
- âÙ
Ö! ,
detÖ
Ù œ + det . â â
! . â â
Õâ â âØ
Õ ã â â âØ
where ‡ means "any number".
Also: deta>Eb œ >8 detaEb (where E is an 8 ‚ 8 matrix)
detˆEX ‰ œ detaEb (transpose property):
+
in E ß elements get flipped across the diagonal, like ”
X
Shortcut for inverting a # ‚ # matrix:
if +.  ,- Á !ß
+
”-
,
.•
"
"
œ +.,”
.
-
,
Þ
+•
,
+
œ”
•
.
,
X
.•
Things You Should Know (But Likely Forgot), page 4 of (
Linear dependence, Spanning and Algebra Issues
A is in the span of ? and @ iff for some +ß , − ‘ß
A œ +?  ,@
The set of all such A is the span of ? and @.
Extended: The span of eB" ß ÞÞÞß B8 f is œA l A œ ! +3 B3 ß +3 − ‘
8
3œ"
A set (list) eB" ß ÞÞÞß B8 f of vectors is "LI" (linearly independend) if the equality
! +3 B œ ! is satisfied only by making all +3 œ !Þ
8
3œ"
Note that no vector in a LI set is a linear combination of the others (i.e. not in the span of
the others).
And specifically, no vector in a LI set can be a multiple of any ofthe others, either.
For 2 vectors: LI simply means that B1 Á const. † B# and B2 Á const. † B1 is sufficient
to demonstrate that eB" ß B# f is LI.
Warning: For 3 or more vectors, one must be more subtle: f9< exampl/
eB# ß "  $B# ß "f is not LI, but none is a multiple of any other, either.
A basis of a span of vectors is the smallest (by number of members) set of vectors that
can be used to span the same space. Note that a basis is not unique.
"
!
"
"
For example, œ” •ß ” • spans ‘# ß as does œ” •ß ”
, etc.
!
"
"
 " •
These bases are inevitably LI, and all have the same number of vectors; this number is
the dimension of the span. For example, ‘# has dimension 2.
Ä Any LI set of vectors in a space, with the same number of vectors as the
dimension is, in fact, a basis.
A time-saving trick:
Linear independence of "sub-vectors" implies LI of the larger vectors, but linear
dependence of sub-vectors proves nothing.
Example:
Ô"×
Ô!×
!
"
!
Ö Ù
Ö"Ù
t" œ ” • ß ?
t# œ ” •
let @t" œ Ö Ùß @t# œ Ö Ù ß ?
"
#
!
"
Õ"Ø
Õ#Ø
t " are the first two elements of @t" , similarly for ?
t # and @t# Þ
The elements of ?
t" ß ?
t # are not proportional, so @t" ß @t# are not proportional, without having to
Clearly ?
check the rest of the elements.
But: the converse is no good: the vectors from the last two elements of @t" ß @t# are
proportional, but @t" and @t# are LI.
Things You Should Know (But Likely Forgot), page 5 of (
Linear operators
0 aBb is a linear function (or operator) iff
0 aB"  B# b œ 0 aB" b  0 aB# b B" ß B# are vectors, or functions
0 a5Bb œ 50 aBb
5−‘
(Note: This implies 0 a!b œ !ÞÑ
t "  6B
t # b œ 50 aB
t " b  60 aB
t #b
Condensed form:
0 a5B
The only type of example in ‘:
0 aBb œ 7Bß 7 − ‘Þ
In ‘8 ß every matrix E with 8 columns represents a linear operator:
t  Ct b œ EB
t  ECt ß and Ea5B
t b œ 5EBÞ
t
E aB
Note: If 0 and 1 are linear then so are 0  1ß 50 (5 − ‘Ñ and so is the composition
t b œ 0 a1aB
t bbÞ
0 ‰ 1 where a0 ‰ 1baB
.
.
.
For example: .B
is a linear operator on functions, as is Š$ .B
#  .B ‹Þ
#
Polar form
aBß Cb Ä a<ß )b (cartesian to polar)
Usually we assume <  !ß ) − Ò !ß #1Ñ or
Ð  1ß 1Ó to maintain uniqueness for all
points other than a!ß !b
(but sometimes it is useful to bend the rule)
<# œ B#  C # ß so < œ ÈB#  C # ß cos ) œ BÎ<ß sin ) œ CÎ< ß
if ) − ˆ  1# ß 1# ‰ ß that is, B  !ß then ) œ tan" ˆ BC ‰ (note the restriction!!!)
B œ < cos )ß C œ < sin ) (the underlined formulas are by far the most useful
ones).
Vectors and geometry
t @t − ‘$ ß their cross-product is
For ?ß
â t
â 3
â
t ‚ @t œ â ?"
?
â
â @"
t4
?#
@#
â
5t â
â
?$ â
â
@$ â
Properties
t ‚ @tl œ l?
t ll@tl sin ) () being the angle from ?
t to @t )
l?
t ¼ a?
t ‚ @tb and @t ¼ a?
t ‚ @tb, the right-hand rule determines the direction.
?
t ‚ @t œ  @t ‚ ?
t,
?
t b ‚ @t œ 5 a?
t ‚ @tb,
t ‚ a@t  A
t b œ a?
t ‚ @tb  a?
t‚A
tb
a5?
?
t @t is l?
t ‚ @tl œ l?
t ll@t l sin ) .
Area of a parallelogram with sides ?ß
Things You Should Know (But Likely Forgot), page 6 of (
Dot product:
t @t − ‘8 then ?
t † @t œ ! ?3 @3 (sum of products of respective coordinates)
If ?ß
8
3"
Properties:
t † @t œ @t † ?ß
t a5?
t b † @t œ 5 a?t † @tbß ?t † a@t  A
t b œ ?t † @t  ?t † A
t
?
t † @t œ l?
t ll@tl cos )ß l?
t l œ È?
t†?
t œ Ë ! ?#3
?
8
t @ß
t A
t À l?
t † a@t ‚ A
t bl
Volume of a parallelepiped with edges ?ß
3œ"
tt
tt
t œ ?†@
tß
t œ l?
t lcos ) œ l?†@
Projection: proj@t ?
component comp@t ?
t t@
@†@
@t l
Inequalities:
¹+t † ,t¹ Ÿ l+t l½,t½ (Cauchy-Schwarz Inequality)
½+t  ,t½ Ÿ l+t l  ½,t½ (Triangle Inequality)
Partial Fractions: For example,
B$ &B# #B
F
G
H
KBL
œ EB  B"
 aB"
 aB"
 IBJ
B# "  aB# "b#
b#
b$
BaB"b$ aB# "b#
Note: (1) Degree of numerator < degree of denominator, i.e., $  "  $  # ‚ #
(2) The powers of factors go from 1 to the original power
To find Eß Fß Gß HÞ IÞ J ß Kß L , clear the fractions, expand the powers and compare
coefficients.
You obtain a system of linear equations, same number of equations as unknowns, and
non-singular (I promise).
B'
E
F
G
œ B"
 aB"
 B$
aB"b# aB$b
b#
On an example:
B  ' œ EaB  "baB  $b  F aB  $b  G aB  "b#
œ EaB#  %B  $b  F aB  $b  G aB#  #B  "b
B  ' œ aE  G bB#  a%E  F  #G bB  a$E  $F  G b
System:
E  G œ !ß %E  F  #G œ "ß $E  $F  G œ '
Solution (any correct method):
E œ  $% ß F œ &# ß G œ $%
Result:
B'
"
"
"
œ  $% † B"
 &# † aB"
 $% † B$
aB"b# aB$b
b#
Some shortcuts:
In the factored form (line 1), set B œ  " to obtain quickly & œ #F , and so on.
You may not get all constants this way, but the ones you do get can speed up the
calculations, say by replacing the unknowns in line 3 by their values.
Taylor Series: 0 aBb œ ! 0 8xa+b aB  +b8 (where 0 is "analytic" and the series is
_
a8b
8œ!
convergent), When + œ !, the series is also caled Maclaurin series.
line 1
line 2
line 3
Things You Should Know (But Likely Forgot), page 7 of (
Handy short forms:
&
const.
cont., cts.
DE
dec., decr.
diff.
dim
equ.
fcn, f'n
FTC
fund.
iff
inc., incr.
LI
MVT
ODE
PDE
pf.
situ.
soln.
Thm
var, Var
"ampersand", meaning "and"
constant
continuous (cont. for "continuity")
differential equation
decreasing
differentiate, differentiable
dimension
equation, equal
function
Fundamental Theorem of Calculus
fundamental
if and only if
increasing
linearly independent (or linear independence)
Mean Value Theorem
ordinary differential equation
partial differential equation
proof
situation
solution
theorem
variance, variable