__The‑Tangent‑Cotangent-Integral-Calculus-Generator__

__A software based calculation device to
help
students practice solving trigonometric integrals__

__Created by David Alderoty © 2011, e-mail David@TechForText.com__

__To contact the author use the above email address
or__

__left click on these
words for a website communication form.__

** **

This website
provides, free of charge, the ** Tangent-Cotangent-Integral-Calculus-Generator**,
which is a software based calculation device, designed to help students practice
solving trigonometric integrals. This software is available for download in
JavaScript, Microsoft Excel, and OpenOffice Calc, formats. In addition,
there is an online version of the

If you want to
download the Tangent-Cotangent-Integral-Calculus-Generator, or want additional
information, scroll all the way down, beneath the online version. Alternatively, you can go to the table of contents of
this website, by left clicking on these words. If you want **to**
go directly to the download section left click on these words. If you want to go directly to the instructions for the
Tangent-Cotangent-Integral-Calculus-Generator, left click on these words.
You should read the instructions before using this software. If you want an online printer friendly version of
the **Tangent-Cotangent-Integral-Calculus-Generator** left click on these
words.

__The Tangent-Cotangent-Integral-Calculus-Generator__

__Downloads and Related Information
for the Tangent‑Cotangent Integral‑Calculus‑Generator__

Step 1) Left click with the mouse on
a blue download *link, for the
__Tangent-Cotangent-Integral-Calculus-Generator__ and a dialog box will open,
with an option to save the file. *Note the download links are located in the
next subsection.

Step 2) Save the file on the Windows Desktop or in the Documents Folder, or anywhere else on your computer, where you can easily locate the file’s icon. Keep in mind that after you downloaded the file, you must locate the file's icon on your computer.

Step 3) The file's icon can be moved after downloading to any location on your computer, with the windows cut and paste function. You should move the icon to a location on your computer where you can easily find it. For most people, the best place to store these files is in the Documents Folder (also called the My Documents in some versions of Windows).

An alternative to the above three
steps, is to download the __Tangent-Cotangent-Integral-Calculus-Generator__
in a zipped folder. To do this left click on a link with the words:
**zipped folder**. With this method, most browsers
display the files icon as soon as the download is completed. Then you can
use the Windows cut and paste function, to place the file anywhere you want on
your computer.

If you need additional information on __downloading,__ left click on
the Google search link, below:

Google Search:[How to download files from the Internet]

__Download Links, for
the Microsoft Excel____
Format__

If you want the above in a zipped folder, left click on these words.

If you want the above in a zipped folder, left click on these words.

If you want the above in a zipped folder left click on these words.

__Download Links, for
the ____OpenOffice Calc Format__

* *

If you do not have Microsoft Excel on your computer, the
best alternative is to use OpenOffice Calc, for the __Tangent-Cotangent-Integral-Calculus-Generator.__** **To do
this, you must first obtain the __FREE
____OpenOffice.org software package, which provides almost the
same__ functionality as Microsoft Excel, Word, PowerPoint and Access.
The OpenOffice.org software package is open-source, and you can download it from
the following website: www.OpenOffice.org

__Download Links, for
the JavaScript____ ____Version__

If you do not have Microsoft Windows
on your computer, you can use the JavaScript version of the __Tangent-Cotangent-Integral-Calculus-Generator.__ The JavaScript version should
work with most modern operating systems, but it was only tested with Windows.

The following two JavaScript versions
are in the ** Web Archive, Single file**, format, and they were
converted to this format with the save function in Internet Explorer. This
file format is different than the conventional HTML, and it has the extension

If you want the above in a zipped folder, left click on these words.

If you want the above in a zipped folder, left click on these words.

__Download Link, for All of
the above, in One Zipped Folder__

__General Description
and Instructions__

__Software and Computer Requirements
for the Tangent‑Cotangent-Integral‑Calculus‑Generator__

All the versions of the
__Tangent-Cotangent-Integral-Calculus-Generator__, are less than 1 MB.
Thus, this software requires very little computer resources, and it should
function well even with the older computers. **However, t he Excel and OpenOffice Calc versions of the
Tangent-Cotangent-Integral-Calculus-Generator, require Microsoft Windows.**

__The Primary Purpose, of
the Tangent‑Cotangent-Integral‑Calculus‑Generator__

The
__Tangent-Cotangent-Integral-Calculus-Generator__ is designed for students
that know the basic method of solving a trigonometric integral, but need
practice to avoid errors, and to increases speed and efficiency with these
calculations. The Tangent-Cotangent-Integral-Calculus-Generator is not
designed to teach trigonometry or calculus, or to explain related mathematical
principles. If you need this type of instruction, you should carry out
Google searches for **VIDEOS** for calculus,
trigonometry, and for solving trigonometric integrals.

__Functionality and Description
of the Tangent‑Cotangent-Integral‑Calculus‑Generator__

__The
Tangent-Cotangent-Integral-Calculus-Generator__ generates twelve math problems,
involving trigonometric integrals, based on two numbers entered by the
user. These numbers are for two angles, designated as B and A, and they
appear on each integral as illustrated in the following example:

Angle B is on the top

Angle A is on the bottom

With this example, the user entered
0.5pi radians (90 degrees) for angle B, and 0.25 pi radians (45 degrees) for
angle A. (Note, the above illustration is an actual copy of an integral
from the Excel version of the
__Tangent-Cotangent-Integral-Calculus-Generator__)

Based on the above illustration, when
angle B is greater than angle A the calculated results will generally be
positive, and when angle A is greater than angle B results will be
negative. This is assuming that **positive angles are entered**, that
are less than 90 degrees, but greater than 0 degrees.

The angles that are entered in the
input boxes of the __Tangent-Cotangent-Integral-Calculus-Generator,__ should
generally be greater than 0 degrees, but less than 90 degrees. If this is
not done, some of the integrals will not have any calculated results. However,
the calculation mechanism in this software can perform calculations, with very
large and very small numbers, which may be positive or negative.

When the user (student) enters numbers, for angles B and A, the software instantaneously calculates the results for the twelve integrals. However, these results are initially concealed from the user.

The student’s job is to try to solve
each integral with pencil and paper, and a calculator for the arithmetic and
trigonometric functions. After attempting to solve each integral, the user
checks his/her calculations, with the results calculated by the software.
To do this there is a pull-down menu, highlighted in pink, ** under each
integral.** The user left clicks on this menu, and moves it down one
level, and the calculated results are displayed. This mechanism is
designed to reveal the results for one integral at a time.

There are input boxes on the top of
the __Tangent-Cotangent-Integral-Calculus-Generator ,__ where the user
enters numbers for angles B and A. The numbers can be entered in either
degrees or radians.

**When radians, are used
with the Tangent-Cotangent-Integral-Calculus-Generator, the angles are
entered in terms of fractions or multiples of pi. Examples are, if you
enter 2, it is NOT two radians; it is 2pi radians, which is equal to 360°.
If you enter 1, it is not one radian it is 1pi radians, which is 180
degrees. If you enter 0.5 it is 0.5pi, which is equal to 90°,
etc.**

I designed the above functionality into the software, because it simplifies the process of entering angles in radians. For example, if you entered one revolution of a circle, or 360 degrees, in radians, using the conventional method, you would have to enter 6.283185307 radians. For one half of a revolution (or 180 degrees) you would enter 3.141592654 radians, and for one quarter of a revolution (or 90 degrees) you would enter 1.570796327 radians, etc. However, with the mechanism I built into this software, when you enter a number in the input box for radians, it is multiplied by pi. This means when 2 is entered it is 2pi radians (or 360 degrees), when 1 is entered it is 1pi radians (or 180 degrees), etc.

Just under the input boxes there are two display boxes that show the angles that were entered by the user in both degrees and radians. Thus, regardless of what the user enters, the display boxes will perform the necessary calculations to display both degrees in radians. For example, if you enter 45°, the display box will show the following:

Just like the display boxes, discussed above, the angles on the integrals are displayed in both pi radians and degrees, as illustrated in the following diagram:

__Scientific Notation and the
Tangent‑Cotangent-Integral-Calculus-Generator__

The
Tangent-Cotangent-Integral-Calculus-Generator may display very large and very
small calculated results in scientific notation, using the letter **e**. (The Excel version uses a capital
**E**) An example of a large number in
this format is:

**4.4218961132108144e+116.**

An example of a very small number displayed in this format is:

**2.5746740266666662e-130**

Extremely small numbers, such as the above might be rounded to zero by the software, because they are generally considered insignificant.

If you want to see very small
numbers, in your calculated results, for whatever reason, there is a control
mechanism on the upper portion of the
__Tangent-Cotangent-Integral-Calculus-Generator__, for setting the number of
decimal places that are displayed. The default is 5 decimal places.
Change this number to 300, which will totally eliminate the rounding
function.

__The Tangent-Cotangent-Integral-Calculus-Generator
is Very Easy to Use, as
Explained in the Following Four Steps.__

**Step 1**) Enter any two numbers you prefer
for angles B and A, in the white input boxes, on the top of the
Tangent-Cotangent-Integral-Calculus-Generator. (It is best to enter numbers that
are less than 90 degrees, but greater than 0 degrees. If this is not done
some of the integrals may not have calculated results.) You can enter the
angles in degrees or radians. *If you want
to enter the angles in radians use the input box on the left, but if you want to
enter the angles in degrees use the input box on the right. It is probably
easier to enter the angles in degrees, if you do not want to deal with
decimals.

With the numbers you enter, the software generates twelve math problems, involving trigonometric integrals, which you are to solve. The software calculates the solution to each of these problems automatically, but the results are initially concealed.

**
**

__*Note When angles
are entered in radians, with this software, they are entered in terms of a
fraction or multiple of PI. Examples are if you enter 2, it is NOT 2
radians, it is 2pi radians,(= 360 degrees); if you enter 1, it is NOT 1 radian,
it is 1pi radians, (= 180 degrees); if you enter 0.5 it is NOT 0.5 radians, it
is 0.5pi radians (= 90 degrees).__

**Step 2) **After entering the numbers in the
white input boxes, scroll down and you will see twelve trigonometric integrals,
which progressively increase in difficulty. Your job is to try to solve
each integral with pencil and paper. To do this, you should have **a list of the formulas needed to solve the
integrals**, and a **scientific
calculator**.

A list of the formulas you need for the calculations is on this website, under the following heading:

__A List of
Formulas to Help You Solve the__

__Math
Exercises Generated by the__

__Trig
Integral Calculus Generator__

**To go
directly to the List of Formulas left click on these
words**

If you do not have a handheld calculator, with trigonometric functions, you can use the calculator provided with the Microsoft Windows operating system. The Windows calculator has a setting for a scientific calculator, and when it is set, the trigonometric functions are displayed on the keypad of the calculator. With the version of Windows that I am working with, (Windows 7), the view menu (on the calculator) has the setting for the scientific calculator.

Another alternative is to use an online scientific calculator. Links to three of these calculators are provided below:

http://www.calculateforfree.com/sci2.html

http://calculator.pro/Scientific_calculator_online.html

**Step 3)
**After attempting to
solve each integral, check your calculations, with the software. To do
this, there is a pink pull-down menu, below each integral, to display calculated
results. A picture of the menu for the ** online and downloadable JavaScript
versions** of the Tangent-Cotangent-Integral-Calculus-Generator is
presented below:

Left click anywhere on the menu, and
it will open. Then scroll down one level, until you see the words: __The
Calculated Result, Displayed Above in Red Type.__ Left click on these
words, and the menu will close. If the above was carried out successfully,
the menu will appear as follows, with the calculated results above it.

The menus in the Excel and OpenOffice Calc versions, of the Tangent-Cotangent-Integral-Calculus-Generator, are slightly different than the above. A picture of this menu is presented below:

To open this menu, it
is necessary to left click once on the pink section, and again on the right
section with the red cross. When this is done, the menu opens, and as with
the JavaScript version, move downward one level, to the words ** The Calculated Result, Displayed
Above in Red Type**. Than Left click
on these words (

**Step 4)** For each practice session, change
the numbers in the white input boxes, if you want a set of twelve integrals that
have calculated results that are different than the integrals in your previous
practice session. **Keep in mind that multiple practice sessions are
usually required to master a mathematical technique.**

__Trigonometric Integral
Formulas, And Related Concepts__

This section provides information
about integral formulas that relate to the
__Tangent-Cotangent-Integral-Calculus-Generator.__ This includes a list
of the formulas, to help you solve the integral problems generated by this
software. __Tangent-Cotangent-Integral-Calculus-Generator__. These formulas are indefinite
integrals. Some of these formulas are frequently illustrated in calculus
classes and in mathematics books.

__The Source: for the Integral
Formulas Listed in This Section__

The integral formulas listed in this section, were
actually calculated by commercially available mathematics software, in
** symbolic notation**. This may sound strange to some of us, but
modern mathematics software, can carry out calculations without numbers, to
generate formulas. For example, I entered:

However, the formulas calculated by the mathematics software are available from conventional reference sources dealing with calculus and trigonometry. Nevertheless, some of the formulas presented in this section might be structured with trigonometric equivalents that are different then the formulas you have in your reference sources, or the formulas you are familiar with.

I simplified some of the formulas calculated by the mathematics software, such as by adding several terms together into one fraction. When this is the case, I present the formula calculated by the software first, followed by the formula with my modifications.

The mathematics software failed to
calculate a constant, which is usually represented by the letter **C,** in all of the sources I have seen with
indefinite integrals. I corrected this with all of the formulas.
However, the **C** is not relevant for the
problems that are generated by the
__Tangent-Cotangent-Integral-Calculus-Generator__.

__Constant at the End of Indefinite
Integrals__

_{}

The above raises the question, why
are constants placed at the end of an indefinite integral, such as The answer to this
question becomes obvious, if you think of the formulas as
anti-derivatives. When you take a derivative constants are calculated as
zero. A simplified example involves taking the derivative of the
following:** ****,** which equals . Now, if we attempt to reverse
the procedure, by taking the anti-derivative of 6X, (without including a **C** at the end of the calculation) we get:
** We did not obtain
the original expression**, when we reverse the procedure. When a
derivative of an expression is taken, information is lost as a result of the
calculation. To represent this potential or actual loss of information the
anti-derivative can be calculated as The

__Creating the Spreadsheet Formulas, with
the Indefinite Integrals, to Create the____ __

__Tangent-Cotangent-Integral-Calculus-Generator__

__ __

The
__Tangent-Cotangent-Integral-Calculus-Generator__, was initially created in
Microsoft Excel, then it was electronically converted to OpenOffice Calc, and
JavaScript. The OpenOffice Calc conversion was carried out with OpenOffice
Calc software, and the conversion to JavaScript was performed with
SpreadsheetConverter.

Thus, I had to create twelve spreadsheet formulas, one for each integral, which would function in Microsoft Excel. To do this, I used the list of indefinite integrals presented below. The basic idea of how this was done, can be illustrated with:

**Note:** The formula I am using to illustrate
the following steps, was not used in the __Tangent-Cotangent-Integral-Calculus-Generator__. The steps presented below are
presented to explain general principles. Many details are left out, to
provide a brief and clear explanation.)

**Step 1)** The indefinite integral was simplified by combining the
fractions, which resulted in

** **

**Step 2)** An integral based on the above, was
written and solved in terms of angles B and A. This resulted in the
following:

** **

**Step 3)** The result was simplified by adding
the two fractions together, which resulted in the following:

** **

**Step 4)** A spreadsheet formula was created
with the above. For the spreadsheet formula, we only need the expression
to the right of the equal sign, illustrated in red type above. For
spreadsheet formulas a **/** is used for
division, and when there is more than one term involved **( )/**. When the above formula is rewritten
with **(
)/** we get:

Step 5) The white input boxes, were renamed as B and A, with the renaming functionality, available in Microsoft Excel. Before these input boxes were renamed, they had default cell designations as names.) With the renaming technique I used, Microsoft Excel can interpret the meaning and value of B and A, by the numbers that are entered in the white input boxes, and calculate with the formula:

Without the above renaming procedure,
Excel cannot interpret the meaning or value of **B** and **A**,
and cannot calculate with any formula written with these letters.

From the above, it is probably obvious that the spreadsheet versions of the integral formulas do NOT look like the conventional mathematics used to solve integrals. However, they are in fact mathematically equivalent, but the notation used to write spreadsheet formulas is different than the conventional way of writing mathematical expressions.

If you want to see the spreadsheet
formulas that I created for the
** Tangent-Cotangent-Integral-Calculus-Generator, **look under each
integral,

__A
List of Formulas to Help You
Solve the Math Exercises Generated by the
Tangent-Cotangent-Integral-Calculus-Generator__

The formulas listed below are presented sequentially, to coincide with the numbering for each problem on the Tangent-Cotangent-Integral-Calculus-Generator.

**Note: some of the formulas listed
below involved natural logarithms, which are represented by ln, or more precisely
ln( ). For example:**

**ln( 4 )=1.386294361
**

*__________________________________*

* Needed For Problem 1*

** **

*__________________________________*

*Needed For Problem* 2

*__________________________________*

*Needed For Problem* 3

** **

*__________________________________*

*Needed For Problem* 4

** **

*__________________________________*

*Needed For Problem* 5

*__________________________________*

*Needed For Problem* 6

** **

*__________________________________*

*Needed For Problem* 7

** **

*__________________________________*

*Needed For Problem* 8

** **

*__________________________________*

*Needed For Problem* 9

*__________________________________*

*Needed For Problem* 10

** **

*__________________________________*

*Needed For Problem* 11

** **

*OR*

** **

*__________________________________*

*Needed For Problem* 12

** **

*OR*

** **

*__________________________________*

__The Design Concepts,
and____
Services Offered by the Author__

__Design Concept to Maximize User‑Friendliness:
For the Website and For
The Tangent-Cotangent-Integral-Calculus-Generator__

I designed the
__Tangent-Cotangent-Integral-Calculus-Generator,__ and this website, in a way
that would maximize efficiency and ease-of-use. The
__Tangent-Cotangent-Integral-Calculus-Generator__ has instructions placed
next to related input cells, and it is laid out with a simple structure, with
large fonts. The website similarly has large fonts, with a similar
structure, and clearly written instructions. For example, the download
links on this website contain precise wording, to prevent confusion, such as:
__If you want the Tangent-Cotangent-Integral-Calculus-Generator in the Excel
format, left click on these words.__

The website is on one long webpage. This provides the convenience of scrolling down or up, from one section to another, and it avoids the unnecessary complexity of pull-down menus, and links to go from one page to another. However, I provided a hyperlink table of contents as an alternative way of navigating the website.

Some of the material on this website is technical. Thus, for an optimum level of comprehension, the reader IDEALLY should have an advanced background in spreadsheets software, coupled with knowledge of programming concepts and calculus. However, I structured each sentence with the goal of minimizing confusion, and maximizing comprehension, for users with varying levels of technical knowledge. In this regard, perfection is never possible, because users come from diverse technical, cultural, and linguistic backgrounds.

I provided many headings and subheadings throughout this text. This allows the user to easily skip material that they find difficult, or uninteresting. The headings and subheadings are also displayed in the table of contents.

__Services Offered by
the Author David Alderoty__

I design and build user-friendly
software based calculation devices for __arithmetic__, __accounting__,
__currency exchange rates__, __algebra__, __trigonometry__,
__correlations__, __calculus__, and databases with built-in calculation
devices. I also create attractive online calculation devices for
websites. I generally make these devices in the Microsoft Excel,
OpenOffice.org, and the JavaScript formats, but I can work with other
spreadsheet formats besides the above. I also create web communication
forms in JavaScript for websites. This includes forms with built-in
calculation devices. **For a list of websites
with calculation devices that I created, left click on these words, or go to the
following website:**** ****www.TechForText.com/Math**** **

I write instructions for the devices I build. I can also write instructions for software and computer devices created by others. In addition, I can write advertising for your websites, products and services.

I can provide the services mentioned
above on a fee-for-service basis, or possibly based on temporary or permanent
employment. If you are interested in my services, and want additional
contact information or more data on the services I offer, you can email me at
**David@TechForText.com**** **or **use a website
communication form, by left clicking on these words.**

For a list of all the services I
offer see **www.TechForText.com** For a list of all my websites
see **www.David100.com**** **My resume is online at: **www.David100.com/R**

My name is David Alderoty, and I am located in the USA, New York City. If you are a great distance from my locality or are in another country, this is not important. I can provide the above services worldwide, because the software and the writing services I offer can be delivered through the Internet to any locality, providing there are no governmental restrictions.

** **

** **

** **

__Introduction to The Hyperlink
Table of Contents of this Website__

This website is more or less laid out
like a book, but it is on one long webpage. You can scroll up or down to
go from one topic to another. However, if you want to examine all the
sections and subsections of this website, use the __hyperlink table of
contents__, below this paragraph. To go to any section or subsection of
this website you can left click on the blue words that relate to the material
you want to read. The yellow highlighted words are sections, and the
un-highlighted words are subsections.

**If you want to go to the top
of the
website, left click on these words.**

__Table of Contents of this
Website Contents__

** **

**The‑Tangent‑Cotangent-Integral-Calculus-Generator** 1

**Downloads and Related Information for
the Tangent‑Cotangent Integral‑Calculus‑Generator**
20

**Three
Steps for Downloading**. 21

**Download Links, for the Microsoft Excel****
Format**
23

**Download Links, for the
****OpenOffice Calc Format** 25

**Download Links, for the
JavaScript**** Version**
26

**Download
Link, for All of the above, in One Zipped Folder** 30

**General Description and
Instructions**. 31

**Software and Computer Requirements for
the Tangent‑Cotangent-Integral‑Calculus‑Generator**
31

**The Primary Purpose, of the
Tangent‑Cotangent-Integral‑Calculus‑Generator**
33

**Functionality and Description of the
Tangent‑Cotangent-Integral‑Calculus‑Generator**
34

**Scientific Notation and the
Tangent‑Cotangent-Integral-Calculus-Generator** 41

**Trigonometric Integral Formulas,
And Related Concepts**. 50

**The
Formulas: Introduction**. 51

**The Source: for the Integral
Formulas Listed in This Section**. 52

**Constant
at the End of Indefinite Integrals**
54

**Creating the Spreadsheet Formulas, with
the Indefinite Integrals, to Create the**. 56

**Tangent-Cotangent-Integral-Calculus-Generator**
56

**The Design Concepts, and
Services Offered by the Author** 66

**Services
Offered by the Author David Alderoty**
69