" 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 !