Unit 1- Sets, Relations and Functions

The language of sets- a Review

1.                  Two ways to describe a set-

a.       Roster form
E.g. {1, 2, 3, 5, 7, 11}- set of prime numbers less than 12

b.      Set builder form
E.g. {x; x is negative odd integer}

2.  
i.    Finite set- A set that contains finite number of elements
ii     Infinite set- A set which has infinite number of elements                        e.g.: {x; x is a prime number}

3. AÌB read as A is subset of B if every element of A is also known as element of B
     i.e. AÌB if  aÌAÌaÌB       
     B is called subset of A

4. Union - AUB={x: xCA or xCB}
5. Intersection- A∩B={x: xCA and CB}
6. Disjoint sets:Þ A∩B=fÞA and B are disjoint sets
7. Complement of a setÞ A’ read as compliment of A where
   A’={x: xÏa and xCU} where u is universal set
8. Venn diagram
   E.g. A={1, 4, 7}
   B={2, 3, 8}

Summary of above

If A={1, 2, 4, 7, 8}
   B={2, 3, 5, 6, 8}
   C={1, 2, 4, 7, 9}
   U={x: 1£x£10}

Then     AÈB={1, 2, 3, 4, 5, 6, 7, 8}
   ApB={2, 8}
   AÌC 
   A’={3, 5, 6, 9, 10} 
   B’={1, 4, 7, 9, 10}

More on sets

1.                  De Morgan’s law
a.       (AÈB)=A’pB’
b.      (ApB)’=A’ÈB’
Or more generally (ⁿ U_i=1 A_i)’= ⁿp_i=1  A ‘_I      
            and (ⁿ p _I=1A _I )’= ⁿ È _I=1A’ _I

 

2.                  A-B={x| xÎA and xÏB} –difference of two sets

3.                  Cartesian product of two sets

AxB={(a,b)|aÎA, bÎB}

(a,b)=(c,d) ⇒a=c, b=d

4.                  Relation – a relation R from a set A to set B is a subset of AxB. A is called the domain of R and B is called co-domain of R.

 We write this as aRB i.e is  a is in relation to B

5.                  types of relation

a.       Reflexive: if aRa for all aÎA
b.      Symmetric: aRb⇒bRa
c.       Transitive: aRb & bRcÞaRc
d.      Equivalance: any relation which satisfies the above

6.                  Functions: A function from A to B written as f: A®B defines a one to one mapping of elements of set A to elements of set B

7.                  Types of functions- let f: A®B

Into fn®if f(A) ÌB

Onto fn®if f(A)=B-------subjective

on one fn if x₁¹, x₁x₂ÎA then  f(x₁) ¹ f(x₂) ®Injective

a function which is both injective and surjective is called bijective

8.                  Composition of f and g®(gof)(a)=g(f(a))

9.                  inverse of o function  f:A®B is f¹: B®A st if f(a)= b then f^-1(b)=a

Unit 2: Trigonometry

1.            Radian-2pradians =360°
       1 radian=180/p=57°16’ approx

2.             Sin q= AB/AO
        cos q=OB/OA
        tan q=AB/OB

3.                  Table of three for different angles

θ	0	30 °= p/6	45 °= p/4	60 °= p/3	90 °= p/2
Sin	0	½	1/√2	√3/2	1
Cos	1	√3/2	1/√2	½	0
Tan	0	1/√3	1	√3	∞

 

4.                  Sign of all trigonometric functions

* Remember All School To College                     

SIN	ALL
TAN	COS

 

5.                  Identities v.       cos(p+x)=-cosx

a.       Sinq=1/cosecq

b.      Tan q= sinq/cosq=1/ cotq

c.       Cosq=1/ secq

d.      Cot q= cosq/sinq

e.       sin²q+cos²q=1

f.        1+tan²q=sec²q

g.       1+cot²q=cosec²q

h.       sin(x+2p)=sin x

i.         cos(x+2p)=cos x

j.        sin(90-x)=cos x

k.      cos(90-x)= sin x

l.         cos(-x)=cos x

m.     sin(-x)=-sin x

n.       tan(-x)=-tan x

o.      cos(90+x)= -sinx

p.      sin(90+x)=cos x

q.      cos(x±y)=cos x cos y± sinx siny

r.        sin(x±y)=sinxcosy ± cosx siny

s.       sin(p-x)=sinx

t.        cos(p-x)=-cos x

u.       sin(p+x)=-sinx

v.       cos(p+x)=-cosx

6.                  All the above identities require some efforts

a.       Multiple and half angles

    i.      Sin2x=2sinx cosx

                                                             ii.      Sin3x=3sinx-4sin³x

                                                            iii.      Cos2x=cos²x- sin²x= 1-2sin²x= 2cos²x-1

                                                           iv.      Cos3x=4cos³x- 3cosx

                                                             v.      1- cos x=2 sin²x/2

                                                           vi.      1+cosx=2cos²x/2

b    sum and products of sines and cosines

   i.      sin x+ siny=2 sinx+y/2cosx-y/2

                                                       ii.      sinx- siny=2sinx-y/2cosx+y/2

                                                      iii.      cosx+ cosy=2cosx+y/2cosx-y/2

                                                     iv.      cosx- cosy=2sinx+y/2siny-x/2

                                                       v.      2sinxcosy=sin(x+y)+sin(x-y)

                                                     vi.      2cosxsiny=sin(x+y)-sin(x-y)

                                                    vii.      2cosxcosy=cos(x+y)+ cos (x-y)

                                                  viii.      2sinxsiny=cos(x-y)-cos(x+y)

 c    for tangent

                i.tan(-x)=-tanx

                                           ii.tan(p/2-x)= cot x

                                          iii.tan(p/2+x)=-cotx

                                         iv.tan(p-x)= -tanx

                                           v.tan2x=2tanx/(1-tanx)

                                         vi.tanx±y=tanx±tany/17tanxtany

                                        vii.tan3x=3tanx-tan³x/1-3tan²x

7.Graphs of trigonometric functions

 

8    Solution of trigonometric functions

a.       Sinq=sina        Þ q=np+(-1)^na
b.   Cosq= cosa     Þq=2np±^a
c.       Tanq=tana       Þq=np+^a

9        to solve the equations of type- acosq +bsinq=c

a. divide throught by Öa²+b²

b. Let tana=b/a Þ sina=b/Öa²+b² and cosa=a/Öa²+b²

c. Substitute sina and cosa

d eqn becomes cos(q-a)=c/Öa²+b²

10.Solution of triangles

a.    a/ sinA=b/sin B= c/sinC

b.      a²=b²+c²-2 bc cosA

c.       b²=a²+c²- 2ac cosB

d.      c²=a²+b²-2bccosC

e.       sinA/2=Ö(s-b)(s-c)/bc; sinB/2=Ö(s-a)(s-c)/ac;

sinC/2=Ö(s-b)(s-a)/ab

f.        cosA/2=Ös(s-a)/bc where s=a+b+c/2

g.       area of triangle=Ös(s-a)(s-b)(s-c)

h.       projection formula

a=bcosC+ccosB

      b= ccosA+ acosC

      c=acosB+bcosA

j     tangent law

                        tanB-C/2=b-c/(b+c)cotA/2

R=a/2sinA=b/2sinB=c/2sinC where R is the radius of circumference of the circle

11.              inverse trigonometric function

a.       properties

i sin-1 (sinq)=q for  -p/2£q£p/2

ii cos-1 cosq=q for 0£q£p

iii tan-1 (tanq)=q for-p/2£θ£p

iv sec-1 (secq)=for 0£q£p/2, p/2<q<p

v cosec-1 (cosecq) for 0<q<p/2, -p/2£q<0

vi cos-1 (cotq)=q for 0<q<p

b          cosec^-1 x=sin^-1  1/x        x>0

                        cot^-1 x= tan^-1 1/x            x>0

                                    = p+ tan^-1 1/x   x<0

                        sec^-1 x= cos^-1  1/x          x.>0

            c          Relations

                  sin^-1x +cos^-1 x= p /2

tan^-1  x +cot^-1 x= p/2

cos^-1  x+ sec^-1 x= p/2

tan^-1 x+ tan^-1 y=tan^-1  (x+y)/(1+xy) if xy<1

2tan^-1 x=sin^-1 2x/1+x² =tan^-1 2x/1-x²= cos^-1 1-x²/1+x²

tan^-1 x-tan^-1 y= tan^-1 x-y/1+xy  if xy>-1

 

            d          Negative inverse

                        sin^-1 (-x) =  -sin^-1 x

                        cos^-1 (-x)= p-cos^-1 x

                        tan^-1 (-x)= -tan^-1 x

                        sec^-1 (-x)= p-sec^-1 x

                        cosec^-1 (-x)=-cosec^-1 x

                        cot^-1 (-x)= p-cot^-1 x