5.5 Multiple-Angle and Product-Sum Formulas

BASIC INFORMATION

In unit 5.5, we learn the double-angle formulas, the power-reducing formulas, and the half-angle formulas.  We will be given all the formulas except for the double-anglge formulas on the test.  We also need to know how to derive all the formulas for the test.

DOUBLE-ANGLE FORMULAS:

HOW TO DERIVE THE DOUBLE-ANGLE FORMULAS:

POWER-REDUCING FORMULAS:

HOW TO DERIVE THE POWER-REDUCING FORMULAS:

HALF-ANGLE FORMULAS:

HOW TO DERIVE THE HALF-ANGLE FORMULAS:

If any of these don't make sense to you, watch this video on how to derive the formulas:

Visit the sites below for additional help:

(Pink) Trig Formula Sheet From Lat Year

Help with the Double-Angle Formulas

Help with Power-Reducing Formula

Help with Half-Angle Formula

Video on How to Derive Half-Angle Formulas

5.4 Sum and Difference Formulas

Background
When working with trigonometric functions, one might come across a function with sin(x+y). However, upon viewing the outputs or the graph or this function, on realizes that you cannot simply distribute and that

     sin(x+y) ≠ sin(x) + sin(y)

This is where Sum and difference formulas come in. We know that sin(x+y) isn't what is above. However, we don't know what it is yet, so we must derive it, or prove it without knowing what the other side of the equation might look like.

Deriving
Starting with sin(x+y), one can draw a picture as follows:

The picture depicts triangle ABC with angle x being added to triangle ACD with angle Y, thus providing side names to work sin(x+y) as 

.

However, this is not a workable equation, as you cannot input anything into the sides as they are right now. We must rewrite until we have all x's and y's instead of sides.

1) Step one rewrites the equation as the sum of two lines as depicted in the picture, allowing for work with smaller triangles that share each of the sides.

2)Step two uses triangle FDC to replace side DF with DCcos(x).

3)Step three uses a property of rectangles to replace side FE with side CB.

4)Step four uses triangle ABC to replace side CB with ACsin(x).

5)Step five uses triangle ACD to replace DC with ADsin(y), and AC with ADcos(y), making all leading terms AD.

6)Step six divides AD from each term, leaving no side lengths and only x's and y's; a formula that can be worked.

Six Sum and Difference Formulas

Deriving each of the formulas this way could be a hassle. Therefor, since we have already come this far, each of the other five that exist can be derived directly from the first one. However, here are all six of them:

Application

These sum and difference formulas are really helpful in trigonometry. They help work through problems with angles smaller than those found on the unit circle. For example,

can be worked out as follows:

Thus giving what we set out to find, an exact measure for an angle that cannot be found on the unit circle.

Chapter 3- Exponents and Logarithms

Chapter three is about two types of transcendental (non algebraic) functions. The two types of covered in this chapter are exponential and logarithmic functions.

As stated in the textbook, an exponential function with base is denoted by

 where , and is any real number.

You can also write and exponential function as
 

where is a nonzero real and is a nonzero positive real number.

People in the real world use exponents to represent things like compound interest, half-lives, populations, bacterial growth, radioactive decay, and many more things.

Basic Principles To Remember About Exponents:

1. To multiply exponential terms with the same base, add the exponents:
   
   
2. To divide exponential terms with the same base, subtract the exponents:
   
   
3. Anything to the zero power equals one:
   
   
4. Anything to the first power equals itself:
   
   
5. A value raised to a negative exponent means:
   
   
6. Fractions in the exponent mean:
   
   
7. Zero to the zero power is indeterminate:
   
   

 For a useful explanation on exponents, visit:
http://www.purplemath.com/modules/expofcns.htm

 Graphs of Exponential Functions and Transformations:


Below are some useful graphs of variations on the graph of from purplemath.com:

You can see how making small changes to the function can yield very different results the same as any other function. Do the transformations seem similar to other functions you learned about in the past math classes?

 In the function , each variable affects the way a graph looks.

• The term affects the vertical stretch and compression.
• The term affects the horizontal stretch and compression
• The term affects the horizontal shift 
• The term affects the vertical shift.

An explanation of domains and ranges of exponential functions can be found on  page 218 of your textbook.

 Also,

• A negative term represents a reflection in the x-axis
• A negative coefficient represents a reflection in the y-axis

One important application of exponents is in finding compound interest. Below is the formula:

An thorough explanation of the formula is on page 222 of your textbooks.
Also, here is an informative site on compound interest:
http://www.purplemath.com/modules/expofcns4.htm

 Logarithmic Functions:

 Here is a definition of a logarithmic function:

For and and ,

  if and only if

The function is called the logarithmic function with base a.




Logarithmic functions, just like their exponential counterparts, are used in the real world to define situations that would otherwise be difficult to represent. Decibels, pH, and the Richter scale all use a logarithmic scale for measurement and representation. Slide rules also use the principles of logarithms to simplify math calculations.

 Properties of Logarithms:

1.   because
2. because
3. because   <= Inverse Properties
4. If  , then   <= One-to-One Property

Remember,

  is equal to , so in some problems it may help to use this property to exponentiate in order  to solve an equation or to simplify a logarithm

 Enough complicated math terms. To explain logarithms simply, look at it this way: a logarithm is the exponent to  which the base must be raised to get the x. The output of a log is an exponent.


For example,

 because when you exponentiate, you get , which we know to be true by the basic principles of exponents.

Here is an example of a logarithmic function graph.

Log (x):


(graphsketch.com is a website you can use to graph functions online to save or share.
 
 Curious, isn't it? This graph represents what value you would have to raise the base (10) to to get the x value (input) of the equation. This is one reason the graph looks unfamiliar. More details about the graphs of logarithms are on page 232 of your textbook.

Natural Logarithms
The Natural Logarithmic Function:

The function defined by

  ,    

is called the natural logarithmic function.

Calculators denote this as LN. Most operations you do to find the natural log of a number will require a calculator. There are, however, some properties you may use to implement logarithms in your math to your advantage.


Properties of Natural Logarithms:

1. because
2. because
3. and
4. If  , then  

 Wanna hear some useless history about famous mathematicians who developed logarithms as a way to simplify some of the tedious calculations they used to have to do while you are in the middle of studying for finals?? Too late- you just did. ;) Jeez! I did not know it took so long to write these blog posts. More to come soon...

Links to other blog posts (see these pages for many examples and more detailed notes):

3-1 Blog Post (Exponential functions and graphs)

3-2 Blog Post (logarithmic functions and graphs)

3-3 Blog Post (Properties of logarithms)

3-4 Blog Post (Exponential and logarithmic equations)

Something weird is going on with the blog pictures--all of them seem out of place.

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