Engineering Mathematics 145 Summary

INVERSE TRIG FUNCTIONS


1 .
Inverse sin The inverse of sin on interval
-


I' is defined as




sinc arcsin arcsince
siny where -- y -* 3 + -x
y y
=




y
= =
=


, ,




X -
Note : arcsin(since) =x for xE /-Yi ? ]

↑ sinCarcsinc) =
x for [-1 ; 1]

Jos
↳ -
-




↑ 2
-
1) ⑧



( -
1i -

2) Theorem :
for x5 (-1 : 1)

· Carcsine) =
-x ) ! -
x
=

arcsinc + C




2 . Inverse cos




y
=
COS y
=
arCOSS The inverse of cos on the interval [0 ; ] is defined as




( 1 ; 4) 3
0-y-M
Tr
-

x
arccos where +1-x-1
y x
cosy
=
=


⑳ ,




- ⑧


(π ; -

1)
Y ↑/
z




-
Note :
arccos(oss)


cos(arcoss) =
=




x
x for



for
x =



E [-1
[0 ; ]

: 1]


(i
Theorem :
for x((-1 : 1)

· Carcoss) =
-x ) -
x
=
arcoss +
C




3 . Inverse tan



y =ARS y
=
arctans The inverse of tan on the interval (*) is defined as




IY *



I
3




I
-
E R
"
X
M arctanse where
-
y= , =

tany
2



- -
Note
:

arctan(tance) =x for E - :* ]

-
-




- tanCarctans) =
x for E R
2
1 2



- M
2
I Theorem :
for x R

Carctance) =
, x2 / c i
=
arctan +
C

,HXPRBOLIC FUNCTIONS
I

sinn= e- COSECRIC =

sinn
I
e -
COSC

sech-Con
e
+

=




2

RIC es -x Core in
tannic=sns=
-




x
ex
sinn
-

+
e




sinhe
y y 20shx y tanns
=
=
=




f I




-
X X




... ↳ >I
-




!. is -
!

-


- -
I
↳




Hyperbolic derivatives /S integrals)
Hyperbolic function function :
identities :

X

d
sinh(x y) sinks coshy costs sinh(2x) sinho= cost o
+
= ·
+
sinhy . =
2 sinks cost a just +
(



cosh(ccty) = coste ·


coshy + sinks .



Sinky cosh-sint" = 1 a cost = sinc


cosh(2x) =
cost +
sinh =
1 + z sink's I-tank : sech a tann= seck




COMPLEX NUMBERS


--a +
bi where a ,
b t R :
Re( -) =
a 3 (m(-) =

b




Addition :


(a +
bi) +
( +
di) =
(a +
c) +
i(b +

d)



Subtraction
:


(a +
bi) -
(c +
di) =
(a -
c) +
i(b -
d)



Multiplication (a bi)( di) (ac bd) i(bc ad)
:
=
+ + + +
-




=(a+
bilk- bd) i
di) =(ac ad)
a
+ -




Division :
where c=0 d = O




We define the compless bi bi
conjugate of number Di If then-E R
=

a -= -
any as
-
-= a + - =
+
a -
.

,

, Geometric representation of compless numbers :




Im
X



Re(-) By (m) -)
iy determined by ordered (C
y) where
:
-= x an pair x
+ = =
,
,




·
Re




it i le me
relationship



...
x




-
=




-
rcOSO



rcoso
between




+
3


irsing
y
= using




=
-=xc+yi3



r(cos6+isinG)
the
length angle' representation is




Standard form of - : - = x +


yi where x =
ReC -1 3y = 1mC-) with cartesian coordinates (sc , y)
,
.




Polar form of --=r(cosotising) where roso
=

Re(-)3 usino =
1mC-) with polar coordinates Cr ol ,
,




1-1
Modulus x
yi x yz
=
:
r
+

of - = + =
- -
=




Argument of - :
arg)-)= arctan ? =
& + 2nk
,
KE .
If Re(-) Co
,
add subtract i to from a The Principle argument of -




is the
unique argument Arg(-) =

8
,
where 0 =(-4in) .




Note :


for two complex numbers -=r(cosx+ isinx) 3 w =
s(cos +
isin) ,
we have that


-w
=

rs(cos(x +
3) +
isin(x +
B)) 3 w =
(cos(a -
B) +

isin(a-5)) .




Sub-note for any -


,
wD ,
-W = -
-w > i =

i


arg)-w)
=



args -) +



arg(w) S arg(n) =

a w!


I heorem De Moirre's formula ne complex r(cosO isino) ,
:




for any ↳ number -= +
,


·


(r(coso+isina))" =
r(cos(no) + isin(nal)

cor) .



-=-1 arg(-r) =



narg(-)



Theorem
:




for a
positive integer n b a non-zero complex number -
let o be an
argument of -




Then all the nith roots of -
are
given by the formula :




isn( ** )
*
-(cos(0 )
+
n
for 90 n-13
·



K =
+
1 2
=
-

, , . . .,