lesson 57 - The fundamental counting principles

Fundamental Counting Principle:
If one item can be selected in m ways, and for each way a second item can be selected in n ways then the two ways can be selected in mn ways.

Example 1: 
How many different two digit numbers are there?

  9       x   10      = 90
^                ^  0,1,2,3,4,5,6,7,8,9     <  these are your choices
1,2,3,4,5,6,7,8,9

Example 2: 
A multiple-choice test has 7 questions, with 4 possible answers for each question. Suppose students answer each question by guessing randomly.

a) How many possible answers are there for each question?
        4    
b) How many different patterns are possible for the answers to the 7 questions on the test?
    4 x 4 x 4 x 4 x 4 x 4 x 4 = 16384 
c) What is the probability that all 7 questions will be answered correctly?
    1/ 16384 

Lesson 50 - 56

Lesson 50-51 : Identity package
Lesson 52- 56:  Practice Tests, Corrections and Chapter Test

Lesson 49 - Identities and double angles

Identities:

Sin( x + y) = sinxcosy + cosxsiny                   Cos ( x + y)= cosxcosy - sinxsiny

Sin( x - y) = sinxcosy - cosxsiny                    Cos ( x - y)= cosxcosy + sinxsiny

Double Angles:

Sin(2x) = Sin (x + x)
= sinxcosx  +  cosxsinx     (like terms)
 = 2sinxcosx  > same as Sin(2x)

cos(2x) = Cos ( x + x)    
= cosxcosx - sinxsinx                 cos^2x  - sin^2
= cos^2x -sin^2x                     = cos^2x - (1 - cos^2x)
>> = 1 - 2sin^2x                        = cos^2x - 1 + cos^2x
                                                 >> = 2cos^2x - 1

Lesson 48 - Conjugate Rule

Rules:
1) Identity/manipulation
2) sin/cos
3) complex fractions
4) Factor + Cancel
5) combine fractions
6) Conjugate Rule

Prove:    sinx  =  1 + cosx
          1 - cosx      sinx


LS                                   =                                     RS 

    sinx    (1 + cosx)       < multiply by the conjugate           
(1-cosx) (1+ cosx)


 sinx (1+cosx)
 1- cos^2x 

sinx (1+cosx)
  sin^2x

1+ cosx                 =                  1+ cosx
  sinx                                          sinx

Is Cos(x + y) = cosx  + cosy
Cos (12 + 37) = Cos(12) + Cos(37)   < Verify Numerically
        Cos 49 = 0.97 + 0.80
         0.66 = 1.78
  NOT EQUAL!!!

You can proof graphically, algebraically, or numerically.

Lesson 43 - 46

Worked on review for midterm, and completed package on November 17th

Lesson 42 - Proofs

Prove:  tanx( tanx + cotx) = sec^2 x

Left side                               =                         Right side
tan^2x  + tanx(cotx)                               
tan^2 + tanx(1/tanx)    
tan^2x  + 1
  sec^2x                                                       sec^2x
                                 LS =  RS

Steps:
1)  Manipulation
2) sin/cos
3) Fraction
4) combine fraction
5) fracor and cancel
6) LS = RS

Prove:

cscx - sinx  = cosx(cotx)

LS                                   =                        RS

1/sinx  -  sinx (sinx)                         cosx( cosx/sinx)
               1    (sinx)                         cos^2x
1  -  sin^2x                                        sinx

     sinx        
 cos^2x
   sinx                    LS=RS

Lesson 41 - Identities

Identities:                                                      Steps:
Cos^2x + sin^2x = 1                                  1)  Manipulation
1+ tan^2x = sec^2x                                    2) Put everything in terms of sin/cos
cotx + 1= csc^2 x                                       3) complex fraction

Simplify:

csc^2x - cot^2x                                   cosx    +  sinx
sinx(1/cosx)                                          secx        cscx
  1   
= tanx                                              (cosx) cosx    +  sinx (sinx)
                                                        (cosx)1/cosx       1/sinx (sinx)
                                                                  cos^2x  +  sin^2x
                                                                           = 1