" How Do We Measure Arc Length ? "

Measuring Arc Length is very easy ! I know it may sound hard because of the word "Arc" but it isn't. To measure Arc Length , all you need to so is remember this formula :

       

Now, using the formula, lets use this circle to find out the Arc Length:

The first step to solving this is finding the angle measure and the radius. In this case the angle measure is 60 degrees. The radius in this case is 1, even though it is not labeled. 

The next stop is to plug in the angle measure and the radius into the formula. 

                                  So now you have :

                 Arc Length =   60  x 2 pi 1

                                       360

you can simplify  60  to  1    and you can also simplify 2 pi 1 to just 2pi by multiplying 2x1. 

                          360      6

                  Now you have :  1  x 2pi    =   2pi  

                                            6                    6

                Simplified you have :  pi 

                                                  3

See ! It was easier than it sounds ! Just remember the formula ! 

Now Try It Yourself :

How Do We Convert Between Radians and Degrees?

Converting between radians and degrees is actually pretty simple.

 RADIANS : One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

To Convert From DEGREES to RADIANS :
      You can use the simple Formula :   π 
                                                            180
This is accurate because since 360° = 2 π 
To find the amount of radians is 90°, we can use this ratio : 90   =  radians
                                                                                           360          2 π    
So you simply have to multiply degrees by 2π and divide by 360. This simplified is π over 180.
Lets use :  105°

You multiply 105 by  π  and get 7π because you simplify 105 π
                              180             12                                    180

To convert from RADIANS to DEGREES: 

      You use the formula:  180  
                                            π 

Lets take 2π  for example, 

You set up the formula to look like this: 

2π *180      The two pi's cancel out and you are left with 360°.
      π

See how simple ? Now Try it Yourself : 

Convert to Radians using:  60° 




Convert to Degrees using :  5 π
                                           12
   

Why The Name Pythagorean Identity is Appropriate

The Pythagorean Identity is similar to that of the Pythagorean Theory.

The Pythagorean Theorem states : a² + b² = c²

The Pythagorean Identity states that: Sin²x+Cos²x=1 . 

The Pythagorean Theorem : 

The Pythagorean Identity : 

          Sin = Opposite              Cos = Adjacent 

                   Hypotenuse                   Hypotenuse

 

              So as you can see the Pythagorean Identity is an appropriate name because when you plug in numbers for sin and cos and square them, the end result will ALWAY BE ONE ! Cool right ? An identity is similar to sameness or oneness. So the fact that you can plug any number in and the end result will always be one makes the name Pythagorean Identity accurate.

                       TRY IT YOURSELF !

 

What is an exponential function ?

The general form of the exponential function can be written :
               f(x) = ab^x
a is the initial value and b is the growth rate.

Let's look at this problem :
      The growth of bacteria in a dish is modeled by the function f(t) = 2^ n . For which value of t is f(t) = 32 ?
                                                                                                             3
The first step is to write out the exponential function .
          32 = 2^ t 
                       3
Next find a common base . To do this you can turn 32 into 2^5 because 2 to the fifth power is 32.

The new function becomes : 2^5 = 2^ t 
                                                          3
Now you solve for x : 5 =  t 
                                         3
You do this by multiplying 3 to every side.
You final answer becomes t = 15.


Now Try It Yourself : 
   A population of rabbits double every 60 days according to the formula P=10(2)^ 1 , where P is the population of rabbits on day t. 
                             60
What is the value of t when the population is 320 ? 

How Do We Solve Exponential Equations ?

Exponential Equations is when a variable is the exponent.
  For Example : 100 = 10^x  
        The missing exponent would be 2 because 10 squared is 100.

Let's try  2^11 = 2^x
Here's a rule to remember : When the Bases are the same, the exponents are equal.
So the missing exponent would be 11.

Lets try something harder.  3^x+4 = 3^7
Since the bases are the same, set up an equation with the exponents.
It looks like this : x+4=7
 You solve regularly for x, which becomes x=3.

Lets try one more problem . 2^x-3 = 4
First you have to create common bases. You can do this by turning 4 into 2^2.
Now the exponential equation becomes 2^x-3 = 2^2
You set up an equation : x-3 = 2, which becomes x=5.

Now try it Yourself : 2^3x+7 = 2^5x-1

How Do We Simplify Complex Fractions ?

Let's get       2       in simplest form
                    x     
                    1
                   2x

  The first step is to rewrite the problem as a division problem, so it would look like this :
                2    ÷   1 
                x        2x

  The next step is to divide as you would any fraction. Keep the first fraction, change the sign from division to multiplication and flip the second fraction.
It should look like this :

               2   *   2x 
               x         1

The next step is to multiply straight across.

                     4x  
                     1x
The final and last step is to cross out anything the numerator and denominator have in common which is X .

  Final Answer :    4   -------> 4 
                            1


Now Try It Yourself : 
                6  
               9x       
                5 
               3x