Trigonometric Identities 

In this chapter, multiple trigonometric identities are discussed. These include reciprocal identities, quotient identities, even/odd identities, co-function identities, and Pythagorean identities. 

Reciprocal Identities:

                   

                   

                       




Quotient Identities:

                    

Even & Odd Functions:
The definition of an even function is that a function is symmetric to the y-axis. For this to be true, the output must always be the same, regardless if the input has the same or opposite sign. Cosine and Secant are the only two functions that fits this definition.

                       
               

           

Co-Function Identities: (*Co-functions of complementary angles are always equal)

                                           

          

           

If you look at the triangle at the right, we can see that that sin(theta) equal to (a/c). Cos [(pi/2)-theta] is also equal to (a/c). Therefore, sin and cosine are referred to as co-function identities. 

Pythagorean Identities:
 

These identities are solely based on the right triangle and the Pythagorean Theorem, hence their name. 

Pythagorean Identity #1

This is how the first identity is derived:
The triangle at the right has a sin(theta) of a/c. Rearranged, (a=csintheta). cos(theta) of b/c. Rearranged, (b=costheta). The Pythagorean Theorem is a² + b² = c²
If we were to substitute sintheta for a and costheta for b, we get the following: 

At this point, we divide by c^2 and we get the first Pythagorean identity listed in the beginning. 

Pythagorean Identity #2

If we divide by cos²ө from the first formula, the cosines will divide out and you will have the inverse of cosine, leaving the second identity:

Pythagorean Identity #3

After dividing the formula above by tangent, we will be left with the inverse and secant over tangent, leaving us with the final Pythagorean identity:

Rules of Logs

Change of Base Formula
                           
                           
Properties of Logarithms

                          

                          
                          
Examples

                         
                         
Using The Properties

Rational Functions:

A rational function can be written as:

f(x) = N(x)

         D(x)

where N(x) and D(x) are polynomials and D(x) is not a zero polynomial. Every polynomial is a rational function.

To graph a rational function you have to find

- x-intercept

- y-intercept

- vertical asymptote

- horizontal asymptote

- the vertical asymptotes are the zeros of the denominator

- the graph has a horizontal asymptote if the degrees of the polynomials are the same. The horizontal asymptote is the ratio of the coefficients. 

To graph the function 

f(x) =          x                

        - x - 2

   

- the x-intercept is (0,0)

- the y-intercept is (0,0)

- the vertical asymptotes are x = -1 and x = 2 

- no horizontal asymptote

Polynomial Functions of Higher Degree

Polynomial Functions and their end-behavior:

When looking at the graphs of Any polynomial function of nth degree, there are two main things to note.
1) The number of x-intercepts= n
2) The number of Relative Extrema = n-1
Relative Extrema: Relative minimums and maximums

End Behavior:
When predicting the end behavior for a polynomial graph, you must look at the leading term/ highest power of x.

• If nth is even, both arrows will point in the same direction.
• If in is odd, the arrows will face in opposite directions.

Example:

Direction of End Behavior:

Right Hand End behavior: To find the direction of the right hand end behavior, you must look at the sign of the leading term.

Left hand end behavior: To find the direction of the left hand end behavior, you look at the exponent of the leading term to see if its arrow will point in the same direction as the right (even) or the opposite direction (odd). 

Example: Find the end behavior of both left and right sides of the polynomial function

The right hand end behavior: The arrow will point down on the right side because the sign on the leading term is negative (-3).

The left hand end behavior will also be pointing down because x is raised to an even power

Polynomial Functions (Completing the Square)

Polynomial Functions:

Defined as in the form:



Restrictions:
- n must be a non negative integer
- Coefficiants must be real
- Powers of x must be whole numbers
- The degree of the polynomial is the greatest value of n (where )
- the domain of a polynomial is always all real numbers


Example of a Polynomial Function:

Converting from Standard Form to Vertex Form (Completing the Square):

Standard Form: 

Vertex Form:     

Start with the function 

Move aside the 15                                  

The goal is to turn            into a perfect square

To do so, we need to add 16. Normally this would be done by adding 16 to both sides, but because it is being shown as a function , we must add 16 and subtract 16 to the same side (equaling 0).

                                                    
                                                             
                                 

Example 1:


                   
          
                 



Example 2: