Lesson 40 - Changing Periods

Solve using no calc.

Sin2x= 0.25
1) Draw SPA
2) Find the angles
3)Divded each side by two      - treat 2x as Ѳ
4) add the period to each side to get other answers

Sin2x = 0.25
   2          2

x= 0.25     x= 2.89
        2               2
= 0.125      =1.45
  + π                +π                   (add the period for the eq'n to get next answers)
= 3.26             = 4.59
x= 0.125, 1.45, 3.26, 4.59

Solve with factoring:

Tan^2 + tanx = 0
    x^2 + x  = 0  ( what its like)  factor!
  tanx(tanx +1) 
tanx = 0          tanx= -1       
         Do SPA                (tan is slope)
x= 0, π               x= 3π /4, 7π /4

Lesson 39- Solving Trig Functions

Equations    vs.      Identites

Equations:  There are 3 different methods to solve an equation
Method One- Solve algerbracially (Draw SPA)
Method Two-  Solve by graphing (use calc.)
put the left side of the eq'n in y1 and the right side of the eq'n in y2
to find what x equals use 2nd trace (=calc) intersection
           example:  eq'n  Sin(x) = 0.35
                                   y1= Sin(x)    y2= 0.35
Method Three- * most used and easiest*
1) Set = 0
2) put eq'n set to 0 into y1
to find what x equals use calc and zeros

Creating general soultions:
a genernal soultion is for stating the point easily when the graph repeats, just add the period and n
ex.
Sin(x) = 0.35     period= 2π
x= 0.36 + 2πn
n is an interger


* Note: for both graphing methods you must create a view screen (graph) and state the domain and range


Example:
3 + Sin2x = 1 - 5sin2x         collect like terms and set to 0!
6sin2x + 2 = 0
x= 1.74 + πn, 2.97 + πn    n is a interger



                 

Lesson 34 - 38

These were practice tests, test and correction days!

Lesson 33- Reciporcal Graphs

Sin = Csc
Cos= Sec
Tan= Cot

Steps:
1)  Draw base graph    ( 1/ sinx)
2) all x-intercepts become VA's
3) Draw 1's
4) Reciporcal heights

*note: tan has its own VA's make sure you dont miss them!!!

Lesson 32- Word Problems

At the seaport, the depth of the water, h metres, at time, t hoursm during a certain day is given by the following formula:

h= 3.2Sin2π(t-3)/12.4 + 3.6


In this equation you know that:
P=12.4
Center line: 3.6
Amp: 3.2 
Range: from 3.6(center line) up 3.2(amp) giving you 6.8 and down to 0.4


Use the calc, to find out the min, and max of the water and what times the water is at a certain height.

Lesson 31

In this lesson we learnt an easier way of graphing periods

Sin and Cos graphs when you have 2π upstairs and a # downstairs,  the number will be your period. 
example:
y= cos 2π/4 Ѳ
P= 4  1/4p= 1
Same goes for Sin as well

For a Tan graph when you have a π upstairs and a number downstairs, the number is your period.
example:
y=tanπ/4 Ѳ
P=4 1/4p= 1

Lesson 30- Period Changes

To find the period of a CosѲ and SinѲ you can use the formula 2π/ k     k: the number in front of x or Ѳ
To find your quarter period you take your period and divide it by 4
The way you find the scale is use the common denominator for the period and the 1/4 period
To find where you start on the graph if there is a phase shift, you put your phase shift over the denominator of the scale

Lesson 29 - y=asin( Ѳ - p) + C

y=cosѲ → 2 cosѲ   2 in front means there is a vertical expansion of 2

y= cosѲ  → cosѲ +1     +1 means the graph will move up  1 on the center line (vertical displacement)

y=1/4sinѲ   ¼ treat as 1’s on your y-axis, instead of 1 and -1 it will be ¼ and -1/4 

y= sin( Ѳ + π/2)    ← π/2  this is called a phase shift, it will move to the left π/2  

y= cos (Ѳ - π/3)     

→ π/3    1/4P = π/2    Phase Shift: = π/3 = 2 π/6  Scale: (common denominator) π/6