Trigonometric Form and deMoivre's Theorem

A complex number can be expressed in three ways.

The relationship between the three can be summarized with a diagram:

Converting from one form to the other involves using the right triangle to relate the sides to the angle.
Sometimes complex numbers are easier to work with in trigonometric form. For example, to multiply, divide, or take a power, it is often easier to convert to trig form and apply:


The third property above can be used to take the nth root of a complex number.

For
For example:

Trig Identities - Day 2

And now it's time to learn how to verify trig identities! - a.k.a. TRIG PROOFS! (insert ominous music here)

To do this you are given two things that are already equal to each other but are still forced to prove that they are equal through the use of tiny baby steps. The end goal of any trig identity verification is to make the left side of the equation and the right side of the equation look the same.

HOWEVER, this must be done with extreme caution, for ONLY ONE SIDE OF THE EQUATION CAN BE CHANGED. If you ever even think of messing around with the other side of the equation, or even going so far as to CROSS-MULTIPLY, you will be promptly cast into the fearsome pit of trigonometric DOOM!!!!

In order to become a trig proof master, one must follow THE RULES (Actually, they're more like guidelines):

1. Don't lose track of what you are trying to prove. Randomly swapping in trig expressions for other trig expressions doesn't usually work. A better plan would be to look at what the end result should be before diving in. Verifying trig identities is not guess and check. (For an example of guess and check, see 'finding the roots of a polynomial.')

2. Work on making the more complicated side less complicated (or rewrite everything in terms of sine and cosine). Making something more complicated than it already is is generally a waste of time.

3. Work from minor functions (csc, sec, cot, tan) to major functions (sin and cos). This makes sense when you realize that sin and cos are a lot less complicated. (See rule #2)

4. Use the common denominator. When you find a common denominator, you can put two fractions together to form one fraction. And, believe it or not, one fraction is in fact less complicated than two fractions (See rule #2). I know, it's weird, but 1 really is less than 2.

5. Use conjugates. Remember how, in the first section of this post, sin θ was equal to 1 / csc θ? Guess what? It still is! And guess what else? In general, sin θ is a lot less complicated than 1 / csc θ. (Rule #2) Is this starting to sound familiar?

6. Use

Did you notice that the left hand side is one fraction and the right hand side is two? And did you notice that they all have a common denominator? Does anyone remember rule #4?

7. Use the Force... just checking to see if anyone was paying any attention.

8. Try not to take the square roots of things or raise them to powers. Remember how we're only supposed to change one side of the equation? If you suddenly make one side exponentially bigger or smaller, what are you supposed to do with the other side to make them match up?

9. Remember:

Last time I checked, trying to put two fractions together by finding a common numerator was not considered to be a very good idea. People who do this are getting way too overexcited about rule #4.

10. A trig proof is like an Oreo cookie. There is always more than one way to verify a trig identity. If your first attempt fails miserably, don't be afraid to go back and try something new!

11. DO NOT CROSS-MULTIPLY! DO NOT ASSUME THE LEFT HAND SIDE EQUALS THE RIGHT HAND SIDE! DO NOT MOVE STUFF FROM THE LEFT SIDE TO THE RIGHT SIDE! DO NOT MOVE STUFF FROM THE RIGHT SIDE TO THE LEFT SIDE! DO NOT PASS GO! DO NOT COLLECT $200! JUST LEAVE THE OTHER SIDE ALONE!

Now let's try asample identity:

Rational Functions!

Rational Functions!

DEFINITION: a rational function is a polynomial divided by a polynomial

Examples:

AND
(corresponding to graphs seen below)

* Rational functions often have guidelines called ASYMPTOTES (shown above in red)

• The first graph (see above) has a vertical asymptote of x=0 and a horizontal asymptote of y=0.
• The second graph has two VAs and one HA.
  
  

To FIND vertical asymptotes: set the denominator equal to zero.

Ex: set x=0 ------ VA (x=0) ---- Domain is All Reals except 0

Ex: set to 0 ------ VA (x= +/- 2) ---- Domain: Reals except +/- 2

Ex: set to 0 ------ NO VA ---- Domain: All Reals

Algebraic Rules for horizontal asymptotes (HA) for the rational function:

1. If n is less than m, then the HA is y=0 (x-axis)
2. If n=m, then the HA is y= a/b
3. If n>m, then there is no HA

Examples:

1. ------ 1<2 y="0

2. ------ 2=2 (n=m) ---- HA: y= 2/3

3. ------ 3>2 (n>m) ---- NO HA

REVIEW: Roots, Y-intercept, x/- Intervals, Symmetry

1. Roots: set numerator equal to zero ---- ---- ---- x= +/- 1

2. Y-intercept: sub in zero ---- = = ------ (0, 1/4)

3. +/- Intervals: breakpoints (roots and discontinuities) {+/-1, +/-2}

VA: x= +/- 2

Then...
Test f(x) in each interval with a number line numbered from (in this case) -2 through 2

4. Symmetry: f(-x) = f(x) ---- Even (about y-axis)

5. VA: x= +/-2 HA: y=1

6. Sketch:

The EFFECT Of Multiplicity of Roots on Graph at VA's

is the same as

VA: ------ x=0, 4 (mult 2) *****

HA: 3=3 ---- 1/1 ---- y=1

Roots: ---- x= -2 and x=2 (mult 2)

Number line...

AND

The Actual Graph:

Common Factors in Numerator and Denominator ---- Hole in the Graph

Graph will be the same shape as except undefined at x=2

On calculator table: for x=0 and 2 --- y will show "error"

** Cancel common factor -- shape of graph -- factors that cancel are the hole -- factors that don't cancel are the VA **

Algebraic Rules For Slant Asymptotes

A function f(x) has a slant asymptote iff ("if and only if") the degree of the numerator is exactly one more than the degree of the denominator.

and n-m=1

To sum things up:

• if n is less than m --- y=0
• n=m --- y=a/b
• n-m=1 --- slant
  

The equation of the slant asymptote is the quotient of the numerator divided by the denominator (disregard any remainder)

EXAMPLE:

You do long division just as you would with normal numbers.

• This comes out with a remainder of 27x-26
• Completely disregard the remainder
• The equation on top is the equation of the slant asymptote!

In the end, you get a slant asymptote of y=2x-7

Lastly...

GIVEN GRAPH, Write The Equation

1.

Roots: (1,0)

• Numerator: (x-1)
• VA: x=2
• Denominator: (x-2)

Solution:

2.

Roots: (-1,0) (1,0)

• Num: (x+1)(x-1)=
• VA: x=2, x=-2
• Denom: (x-2)(x+2)=

Solution:

3.

Roots: (0,0) mult 3

• Num:
• VA: x=3, x= -3
• Denom: (x-3)(x+3)=

Solution:

** Sometimes you just have to fool around with it in your calculator- plug in a negative or an integer.

** Just for the record, when a graph "bounces" at a root- the muliplicity of that root is 2.

And now for...

REVIEW PROBLEMS

1. Give the equation of the horizontal asymptote of .

2. Give the equation of the slant asymptote of .

3. Find the equations of all vertical asymptotes of .

4. Write the equation of a rational function which has a horizontal asymptote of y=3/2 and y-intercept of -2.

5. Write the equation of a rational function which has roots 0 and -5 and whose vertical asymptotes are x=+/-2.

6. Given the graph, find the equation of the rational function.

7. Sketch the rational function .

8. Find the correct equation for the graph below that has a SA of y= 1/2x + 5/4 and a VA of x= 1/2

9. Given the equation, , find the horizontal and vertical asymptotes.

10. For the following equation, , find the roots, the y-intercept, vertical asymptotes, horizontal asymptote, +/- intervals, symmetry, and domain.



---------------------------------------------------- Answers below --------------------

Answers:

1. HA: y= -4

2. SA: y= 2x+4

3. VA: x= -3/2 and x=2

4.

5.

6.

7.

8.

9.

VA: x= 5/2 and x= -1

HA: y= 5

10.

Roots:+/-i (no real roots)

Y-int: (0,1)

VA: x= +/- 1

HA: y=1

Intervals: - (less than -1), + (-1-0), + (0-1), - (greater than 1)

Symmetry: y-axis (x=0)

D: (-infinity, -1)(-1,1)(1, infinity)