Aim: How do we evaluate inverse trigonometric relations and functions?

We Usually know some of the regular trigonometric functions such as; Sin(x) , Cos(x), Tan(x) and so forth .. Other functions like; Cot(x), Sec(x), and Csc(x).

We can solve for the measure of an angle if we know the value of a trig function. By doing this, we use the inverse functions;
    ArcSin x -> Sin ^ -1 x
    ArcCos x -> Cos ^ -1 x
    ArcTan x -> Tan ^ -1 x
    ArcSec x -> Sec ^ -1 x
    ArcCsc x -> Csc ^ -1 x
    ArcCot x -> Cot ^ -1 x

For Example ; ArcSin (1/2) is equal to 30 Degrees

      The Value of Arc Sin (1/2) + Arc Tan (1) is
75 Degrees .
     ( 45 + 30 = 75 )

We have a simple format when converting decimal degrees to minutes.
     We first multiply the decimal portion by 60.
For Example: Write 12.75 in degree/minute format.
    .75 * 60 = 45 '
12.75 Degrees = 12 Degrees 45 '

How do we convert between radians and degrees?

Definitions:
     Radian: One Radian is an angle formed at the center of a circle by an arc that is equal in length to the radius of the circle.

Converting Degrees to Radians:
360 Degrees is equal to 2π(pi)

To Find out how many radians are in 90 Degrees we need to set up a ratio.
         90  = Radians
        360        2π
We then multiply degrees by 2π and divide by 360, simplified is:  π  
                                       180

  90   π = 1 π = π
 180         2       2

Converting from Radians to Degrees:

Formula:  Multiply 180

 π    

For Example:

π * 180

6     π  

Cross out the π's because they cancel out each other. 

So you just divide 180 by 6 and that will equal to 30 degrees

Lets Try Some Problems:

Convert to Radians:

1. 60 π  2. 45 π 

   180       180

= 1/3π = π/3    =1/4π = π/4

Convert to Degrees:

     1. 2π*180      = 360 Degrees

             π

     2. 5π * 180    = 900/12 = 75 Degrees

          12    π

"Explain why the name pythagorean identity is appropriate"

Question of the Day: "Explain why the name pythagorean identity is appropriate"

Well for starters, the formula for the pythagorean theorem (used for a triangle) is a² +b² =c² ; where a and b are the sides of a triangle and c is the hypothenuse. It allows people to identify the sides of a Right triangle.





Not only do we use the pythagorean identity in Algebra, we use the pythagorean identity in Algebra II/Trig. How is this possible ?

   We create a circle like the figure shown on the left, we use one side of the diameter and create a triangle from the "x and y axis" (of the cicle). The Blue line can be identified as the radius of

the circle or as the hypothenuse of the right triangle. We can say, that the radius of the circle is 1. But how does the Pythagorean apply to Trig.? Its simple, let us express sin and cos to the triangle; 

 

 




 

 

sin(ɵ) is expressed as opposite / (over) hypotenuse which is ; y / 1 = y

and

cos(ɵ) is expressed as adjacent / hypotenuse which is equal to x / 1 = x

The Pythagorean Identity is an appropriate name because we can use the formula; a² +b² =c² to say that sin²(ɵ) + cos²(ɵ) = 1 *our trig. formula* The Pythagorean Theorem has simply developed to help us excell in Algebra II/Trig. We can now identify cot, sec, csc, and tan by using the Pythagorean Identity's help.

Example:

sec(ɵ) = 1 / cos(ɵ)

csc(ɵ) = 1 / sin(ɵ)

cot(ɵ) = 1 / tan(ɵ) which is equal to cos(ɵ) / sin(ɵ)

How do we perform compositions of functions?

Compositions of Functions:
     Example: If f(x) = x^2 and g(x) = x+6
Find f(g(4)).
   g(4) = 4+6 = 10
f(10) = 10^2 = 100 .

Compositions with Variables:
   Example: If f(x) = x^2 and g(x) = x+4
Find f(x+4)
    f(x+4) = (x+4)^2
(x+4)(x+4)            (Distribute)
      x^2 + 4x + 4x + 16
x^2 + 8x + 16 .

The term "composition of functions" (or "composite function") refers
to the combining of functions in a manner where the output from one function
becomes the input for the next function.

In math terms, the range (the y-value answers) of one function becomes
the domain (the x-values) of the next function.

The notation used for composition is: and is read "f composed with g of x" or "f of g of x".

(Regentsprep.org)

How do we identify one-to-one fuctions?

A function is said to be one-to-one, if for each element of the range, there is a unique element of the domain. (for every y there is one and only one value x). if whenever
f (a) = f (b) then a = b.

Lets try out these problems,

1. {(3,2), (1,7) , (0,2)} this is a function but it is not a one-to-one function.
2. {(2,1) , (4,2) , (1,4)} this is a function and it is also a one-to-one function.

(regentsprep.org)

"One-to-One"

NOT "One-to-One"

How do we use inverse variation?

First, lets list ALL the interger dimensions for rectangles whose area is 24
                            L           |         W
                               1                 24
                               2                 12
                               3                  8
                               4                  6
                               6                  4
                               8                  3
                              12                 2
                              24                 1
This relationship is called an Inverse Variation.
We say that x varies inversely with y.
       xy = 24                       when x goes up, y goes down.
       y = 24/x                      when y goes up, x goes down.

If x varies inversely as y and x = 12 when y = 3, what is the value of x when y = 9?
                     xy = k                            
12*3 = k
k would equal 26

x*9 = 36
  9        9
x would equal 4

How do we solve exponential fuctions?

Well we already know how to solve exponential equations, or at least we get the hang of it.
     Our new challege is to solve exponential functions.

The general form of the exponential fuction can be written:
     f(x) = ab^x
this is where a is the initial value, and b is the growth rate.
     w(t) = 100 * 2^t

Lets try this problem;
     A population of rabbits doubles every 60 days according to the formula P = 10(2)^t/60 , where P is the population of rabbits on day t, what is the value of t when the population is 320?

a ; would be the initial population, where b would be the growth rate.
          320  = 10(2)^t/60
           10            10

          32 = 2^t/60
         2^5 = 2^t/60
       60*5 - t/60*60    (cancels out)
t would equal 300

How do we solve exponential equations?

Well usually we know how to solve problems like:
          x = 3^4
which will equal 81.

We know how to also solve problems when the unknown is alone:
          (3)^3 = (x^1/3)^3
which will equal; x = 3^3 = 27

We know how to solve problems where even the base is unknown:
          4 = x^2
which will equal to -2,2.

What we didn't know is this;
          100 = 10^x , but this problem is easy, x would have to equal 2 in order for this to be true.
this problem is where the unknown is actually the exponent.

          Solve for x;
2^11 = 2^x
which will equal 11.
          When the bases are the same, the exponents have to be equal as well.

Solve for x:
          2^x-3 = 4
        2^x-3 = 2^2
            x-3 = 2
            +3=+3
    _____________
Final Answer, x=5

Aim: How do we multiply and divide rational expressions ?

Multiplying and dividing may seem difficult, but when using them to solve rational expressions, it can be a lot easier. First, we start out by factoring out all of the numerators and the denominators so we can make things much easier for us. Then we get to cancel out all of the common factors to simplify the expressions. Lastly we have to multiply the remaining factors in the numerator & the denominator. We can find out the answer with these simple steps.
   Problem:

2x +10x + 12x          x + 7 
____________  *     _____      
    
    2x + 10x               x² − 9

                        ↓

2x(x +5x +6)       x + 7 
__________  *    ________            (Factor it Out)
2x(x + 5)          (x+3)(x−3)

                   ↓

 2x(x+2)(x+3)           x + 7 
____________   *    _______             (Cross Out)
2x(x + 5)              (x+3)(x−3)
                   ↓

 (x+2)          (x +7)         x² +9x + 14    (Multiply Remaining Factors)
______   *   ______  =   ___________

 (x+5)          (x−3)          x² +2x − 15

  
                   ↓

x² +9x + 14
__________                (x cannot equal; 0, 3,−3,−5)
x²  +2x − 15


DIVIDING:
   First you start out by using KCF (Keep, Change, Flip) which is the multiplication of reciprocal.
  Problem : 
x + 7                   x² + 9x + 14                        x + 7            3x² - 9x
____        /      ___________        KCF        _____    *   __________
x² − 9                  3x² − 9x                             x² − 9       x² + 9x + 14

Step 2: Factor out everything that needs to be factored.
     x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 3: Cancel out the common factors

    x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 4: Multiply the remaining factors of the numerator & denominator.

      3x                        3x
_________    *    ____________
(x+2)(x+3)           x² + 5X + 6

Final Answer

    3x
__________     (x cannot equal;  0,−2,3,−7,−3)
x² + 5X + 6


http://www.mathwarehouse.com/algebra/rational-expression/how-to-multiply-divide-rational-expressions.php

Aim: How do we solve problems with the rational expression operations?

For starters, Rational Expressions can be defined as fractions that have variables in their denominator.

Multiplying Rational Expressions: We should begin by searching for things that can be eliminated otherwise known as factoring it out. Once you have that done, you can start to divide out the common factors and simplify the expression. For Example ;



 -> You look to see if you can factor anything out.
Nothing can be factored out just yet.
 -> Divide the common factors.

= 

-> Simplify.


Final Answer will be