Fundamental Identities

If an equation contains one or more variables and is valid for all replacement values of the variables for which both sides of the equation are defined, then the equation is known as an identity.

The Unit Circle

The Unit Circle 

 

What is the Unit Circle?

The unit circle is a circle whose center is at the origin with a radius of one. Because the radius is 1, you can directly measure sine and cosine. If a point on the circle is on the terminal side of an angle in standard position, then the sine of such an angle is the y-coordinate of the point, and the cosine of the angle is the x-coordinate of the point.
 

 

The Pythagorean Theorem states that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:

 

x²+ y² = 1²

But 1² is just 1, so:

x²+ y² = 1
(the equation of the unit circle)

Using Coordinates to Find Trigonometric Functions

The coordinates x and y are two functions of the real variable theta. You can use the coordinates to define the six trigonometric functions of theta.

The point of the unit circle is to make math easier and neater. For example, in the unit circle, you have, for any angle theta, the trig values for sine and cosine are sin(θ) = yand cos(θ) = x. Working from this, you can take the fact that the tangent is defined as being tan(θ) = y/x, and then substitute for x and y to easily prove that the value of tan(θ)also must be equal to the ratio sin(θ)/cos(θ).

Special Points of Interest on the Unit Circle

It is very important to memorize the sine and cosine of the angles created by special right triangles for future use.

 

 

Here is a video on how to easily remember the Unit Circle-

For further help, please watch these videos-
http://www.youtube.com/watch?v=ZffZvSH285c
http://www.youtube.com/watch?v=DIGoK51u0KQ
http://www.youtube.com/watch?v=3GgO7Q_kg8Q

 

 -Kenji Johnston

 



 



Questions/Muddy points for Chapter 2

Please comment on this post using the link below.

Write a question or concern you currently have regarding any of the material we've covered so far. 

Rational Functions

For graphing rational functions we need to follow these steps: 

1)  Find any intercepts, if there are any.  (Remember to find the y-intercept with f(0) and x intercepts by setting the numerator equal to zero).

2)  Find the vertical asymptotes by setting the denominator equal to zero and solving.

3)  Find the horizontal asymptote.

4)  The vertical asymptotes will divide the number line into regions.  In each region graph at least one point in each region.  This point will tell us whether the graph will be above or below the horizontal asymptote and if we need to we should get several points to determine the general shape of the graph.

5)   Sketch the graph.

Example:

Sketch the graph of the following function.

                                                            

This time notice that if we were to plug in  into the denominator we would get division by zero.  This means there will not be a y-intercept for this graph.  We have however, managed to find a vertical asymptote already.

Now, let’s see if we’ve got x-intercepts.

                                     

So, we’ve got two of them.

We’ve got one vertical asymptote, but there may be more so let’s go through the process and see.

                           

So, we’ve got two again and the three regions that we’ve got are ,  and .

Next, the largest exponent in both the numerator and denominator is 2 so by the fact there will be a horizontal asymptote at the line,

                                                                 

Now, one of the x-intercepts is in the far left region so we don’t need any points there.  The other x-intercept is in the middle region.  So, we’ll need a point in the far right region and as noted in the previous example we will want to get a couple more points in the middle region to completely determine its behavior.

                                          

Here is the sketch for this function.

Definition of Vertical and Horizontal Asymptotes

1. The line x = a is a vertical asymptote of the graph of f if       or

     - as x  a, either from the right or from the left.

2. The line y = b is a horizontal asymptote of the graph of f if    as    

      or  -.

The line x = 0 is a vertical asymptote of the graph of f, shown below.

The graph of f also has a horizontal asymptote - the line y = 0.  This means the values of f(x) = 1/x approach zero as x increases or decreases without bound.

    as   -                                                             as   

       approaches 0 as x                                                                                          approaches 0 as x

       decreases without bound.                                                                                     increases without bound.

  Asymptotes of a Rational Function

Let f be the rational function

 

           

 where N(x) and D(x) have no common factors.

1. The graph of f has vertical asymptotes at the zeros of D(x).

2. The graph of f has at most one horizontal asymptote determined by comparing the degrees

    of N(x) and D(x).

            a. If n > m, the line y = 0 (the x-axis) is a horizontal asymptote.

            b. If n = m, the line y =   is a horizontal asymptote.

            c. If n > m, the graph of f has no horizontal asymptote.  

 

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