Identities

Identity- An equation that is always true

Ex: 2x+5 = 2(x-3) + 11 <---- No matter what you put in for X the sides will always be equal

Identities are used for substitution

Reciprocal Identities:

Quotient Identities:






Pythagorean Identities:

Pythagorean Identities can be derived from other equations. Such as-









Using those two factors we can derive the first pythagorean identity by using the pythagoean theorem and finding a new identity that we can use to substitute.

Plug in the factors

distribute the square

factor out a c^2

divide by c^2 to get the first Pythagorean identity

The other two identities can be derived in a similar way. The three Pythagorean identities are:

Chapter 4, Section 1

Chapter 4, Section 1
Radian Degree and Measure

• The word trigonometry is a word derived from Greek language meaning "measurement of triangles." It deals with the relationships among the sides and angles of triangles
  
  
• An angle is determined by rotating a ray about its endpoint
  

• The starting position of a ray is the initial side of the angle, and the position of the ray after rotation is the terminal side (shown below)
  
  


• Positive angles are generated by a counterclockwise rotation
• Negative angles are genereated by a clockwise rotation

Radian Measure

• One radian is the measure of a central angle (theta) that intercepts an arc (s) equal in length to the radius (r) of the circle


• Because the circumference of a circle is 2PIEr, it follows that a central angles of one full counterclockwise revolution corresponds to an arc length of s = 2PIEr. Therefore, 2PIE radians corresponds to 360, PIE radians corresponds to 180, and PIE over 2 radians corresponds to 90. Because 2PIE = about 6.28, there are just over six radius lengths in a full circle.
  
• In general, the radian measure of a central angle theta is obtained by dividing the arc length (s) by r. Therefore, s/r= theta, where theta is measured in radians.
  




• Because the radian measure of an angle of one full revolution is 2PIE, you can reason the following:

1/2 revolution = 2PIE/2 = PIE radians

1/4 revolution = 2PIE/4 = PIE/2 radians

1/6 revolution = 2PIE/6 = PIE/3 radians

• There are four quadrants in a coordinate system and are numbered I, II, III, and IV

Coterminal Angles

• Two angles are coterminal if they have the same initial and terminal sides

Example: The angles 0 and 2PIE are coterminal, as are the angles PIE/6 and 13PIE/6

• You can find that an angle is coterminal to a given angle theta by adding or subtracting 2PIE (one revolution)

Example: For the positive angle 13PIE/6, subtract 2PIE to obtain a coterminal angle

13PIE/6 - 2PIE = PIE/6

Degree Measure

• A second way to measure angles is in terms of degrees
  
• A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by PIE radians/180 degrees
   
2. To convert radians to degrees, multiply radians by 180 degrees/PIE radians

*PIE radians = 180 degrees

http://www.google.com/imgres?imgurl=http://rml3.com/a20p/images/trig_radians.gif&imgrefurl=http://rml3.com/a20p/trig.htm&usg=__JrvroDr4f3OQmQztOFRpKihX1YQ=&h=281&w=383&sz=11&hl=en&start=4&zoom=1&um=1&itbs=1&tbnid=niDqWHZL4RTiJM:&tbnh=90&tbnw=123&prev=/images%3Fq%3Dradians%2Bin%2Ba%2Bcircle%26um%3D1%26hl%3Den%26sa%3DN%26rlz%3D1R2GGLL_en%26tbs%3Disch:1

Chapter 4, Section 2

Trigonometric Functions: The Unit Circle

4.2: Trigonometry

Okay, so when theres like a right triangle, its got 6 different trigonometric things, these are called:

Sin (oposite leg / hypotenuse)

Cosine (adjacent leg / hypotenuse)

Tangent (opposite leg / adjacent leg)
(also simply sin / cos)

Each of these has a reciprocal:

cosecant = 1 / sin

Secant = 1 / cosine

Cotangent = 1 / tangent

These are reciprocals NOT inverses

On the unit circle, the sin = y value and the cos = x value

a trick to remeber: every 15 degrees = pie / 12

examples:

• 30 degrees = 2 pi / 12 = pi / 6
• 45 degrees = 3 pi / 12 = pi / 4
• 90 degrees = 6 pi / 12 = pi / 2

2.6: Rational Functions

Rational Functions


A rational function is a function that can be written as

Where N(x) and D(x) are polynomials.

• Example of a rational function:

A function likeis considered a rational function, because it can be rewritten as

*NOTE: When you divide by a fraction, the answer gets larger.

__________________________________________________________________

Vertical Asymptotes

Consider the equation .

The denominator cannot be 0, so x ≠ 3.

 Notice that the graph avoids x = 3, and



There can be more than one vertical asymptote, as is demonstrated in the equation, because x = ± 1.

 

__________________________________________________________________

Horizontal Asymptotes

Unlike vertical asymptotes, there can only be one horizontal asymptote.

In order to determine what the horizontal asymptote is

• Pretend that x is a large number like 1,000,000,000
• Consider the highest degree in the numerator and denominator

Because x is so large, everything but the highest degree polynomial is irrelevant. There are three different horizontal asymptote scenarios.

1. Higher degree polynomial in the denominator

 

 Look only at the highest degree polynomials, 3x and 2x²

, so the H.A. is 0.

2. Highest degree polynomial same in both numerator and denominator

Look at only the highest degree polynomials, 3x² and 2x². The x²s cancel each other out, and , so the H.A. is 3/2.

3. Higher degree polynomial in the numerator

It does not level off, so and there is NO H.A.

 Still confused? CLICK HERE to watch a video explaining rational function asymptotes.

Section 2.4 Complex Numbers

A complex number is the combination of a real number

and an imaginary number.

Adding Complex Numbers

Subtracting Complex Numbers

Multiplying Complex Numbers

Dividing Complex Numbers

Conjugate
use when there is an imaginary number in the denominator.

2+3i    the conjugate is 2-3i



Solutions to review

Answer key

Don't forget to check out Assignment 4 on your sheet for a nice overview of the kinds of problems you should be able to do.

Well, here ya go.

2.3: Real Zeros of Polynomial Functions

To find the real zeros of polynomial functions, one can use both long division as well as synthetic division.

Here is an example of how to use long division:

Divide f(x) = 6x^3 - 19x^2 + 16x - 4 by x - 2.

Now you know that both x - 2, 2x - 1, and 3x - 2 are the factors of f(x) (2x - 1 and 3x - 2 being the factors of 6x^2 - 7x + 2). You then set these values equal to zero and solve for x, giving you the zeros of the f(x) equation.

Here is a video that will better explain long division of polynomials:

http://www.khanacademy.org/video/polynomial-division?playlist=ck12.org%20Algebra%201%20Examples

Here is a video that will go through the process of synthetic division of polynomials.

http://www.youtube.com/watch?v=bZoMz1Cy1T4

Note 1: The end of the video about synthetic division also goes over the remainder theorem at the end. The remainder theorem reads as follows:

If a polynomial f(x) is divided by x - k, the remainder is r = f(k).

Note 2: When using synthetic division with polynomials that have 0 as some of their coefficients, you must include the 0 in the problem in order to ensure you get the right answer.

The Rational Zero Test

If the polynomial f(x) = ax^n + ax^n-1 + ... + ax^2 + ax + a has integer coefficients, every rational zero of f has the form:

rational zero = p/q

where p and q have no common factors other than 1, p is a factor of the constant term, and q is a factor of the leading coefficient a.

For your enjoyment:

PhD a.k.a. A Higher Degree

The graph of a polynomial has no breaks, holes, or gaps.

Pretending like i didnt use paint and my lines were actually acurate, we can see what i mean.  The first graph is good cause it has no breaks, the second is not a polynomial because it has breaks.  The third is "theoretically" good cause it has no sharp turns unlike the 4th, which has many sharp turns.

THE LEADING COEFFICIENT

Now, we examine End Behavior.  End behavior is where the right and left ends of a polynomial graph points too.  In the first of the two graphs above, the two ends point in different directions, this lets us know that the leading coefficient is negative.  In the second graph, the two ends are pointed in the same direction, thus, the leading coefficient is positive.

The notation we use to show the first graph's end behavior is:

for the left end and

               Those last 6 lines took me just about 45 minutes

Now we can apply the things we have learned to some equations.

Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right. Like this:

It can be shown that for a polynomial function f of degree n, the following statements are true.

1.  The graph of f has at most n real zeros.

2.  The function f has at most n-1 relative extrema (relative minimums or maximums).