Trigonometric Functions via TI-92

This is an expended abstract of the notes for the workshop on how to use TI-92 on August 24-25, 1996 at the Department of Mathematics of the University of Central Florida.

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Since this is the second of a series of ten lectures, I decided to concentrate on presenting the material to achieve the following two goals:

1. to get familiar with the basic function keys of TI-92, more precisely, to know how (and, more importantly, when) to use the following keys on TI-92's keyboard:

, , , , , , , ,

2 . to learn, in particular, how to manipulate trigonometric functions on TI-92, more precisely, to practise on using the two programs tExpand and tCollect, and, of course, the trigonometric function keys:

Section 1. The Three Main trigonometric Functions Built in TI-92

sin(x), cos(x), and tan(x) (and their inverses) are the three trigonometric functions built in TI-92. The keys for these three functions are located on the right-hand side of the TI-92 keyboard.

Pi is located on a key near tan( ) as a second function (use the second function key ):

The angles can be measured in Radian or in Degree. Use mode to choose between the two.

The other trigonometric functions (like sec( ), csc( ), cot( )) can be defined by simple macros. See the section on How to Define Other trigonometric Functions.

The use of these trigonometric functions on a TI-92 is almost identical to that on any ordinary scientific calculator except that, on TI-92, we can do symbolic calculation! Let us see some examples.

Example 1.1. Type sin( pi/2) followed by to produce 1, and type cos(pi) and then to produce -1, etc.

Example 1.2. Type (by using the second function key on top of the right-hand side of the keyboard) $sin$^-1(1) followed by to produce pi/2. Press key near the ON key to get an approximate value 1.5708.

Example 1.3. Let us try some composed functions. Type sin($sin$^-1(x)) then to produce x. How about typing $sin$^-1(sin(x)) followed by ? Surprise!(Follow this link to the end to see the answer.)

Example 1.4. Guess what will happen when we type tan($sin$^-1(x)) and ?
TI-92 returns the right answer (in pretty type):
x/(1-x^2)^{1/2}.

Some exercises:
Exercise 1.1. $cos-1(1/2)$.
Exercise 1.2. $cos-1$(cos(x)).
Exercise 1.3. sin($sin-1$(x)).
Exercise 1.4. sin($cos-1(x)$).
Exercise 1.5. tan($cos-1(x)$).
Exercise 1.6. sin(2 $cos-1(x)$). (Hint: You may need to read section 3 before you try to solve this problem.)
Exercise 1.7. tan(x)cos(x).

Section 2. The Two Commands for Manipulation of the Trigonometric Functions

There are only two built-in commands ( programs, or macros) in TI-92 for dealing with the trigonometric functions. One is called tExpand, the other, tCollect. For best results, use angles in Radian mode.

There are two ways to access these two commands (actually, the same for all programs in TI-92): the first way is by typing the name of the command; the second is by using the function key (on the left-hand side of the keyboard) F2(algebra) followed by choosing 9 (trig), then the two commands can be chosen from the menu. I prefer the first method. I only use the function F2 to find out what's available.

tExpand( expression ) returns an expression in which sin( ) and cos( ) of multiple angles, angle sums, and angle differences are expanded into those of single angles. In other words, double (or multiple) angle formulas and addition formulas are used to expand the given expression.

Let us look at some examples.

Example 2.1. tExpand( sin(2 x) ) followed by will produce 2 sin(x) cos(x)

Example 2.2. tExpand( cos(x-y) ) followed by will produce cos(x) cos(y) + sin(x) sin(y)

tCollect( expression ) returns an expression in which products (or integer powers) of sin( ) and cos( ) are converted to a linear combination of sin( ) and cos( ) of multiple angles, angle sums (or differences). In other words, angles are collected. Let us look at some examples.

Example 2.3. tCollect( 2 Cos(x)^2 ) followed by will produce Cos(2 x) + 1

Example 2.4. tCollect( 2 Sin(x) Cos(y) ) followed by will produce Sin(x+y) + Sin(x-y)

tExpand and tCollect are almost inverses to each other.

Example 2.5. tCollect(tExpand(sin(2 x))) followed by
will simply return sin(2 x).
But typing the following
tCollect(tExpand(sin(2 x)sin(x))) followed by
will not return sin(2 x)sin(x). It is in this sense that tExpand and tCollect are not exactly inverse to each other.

Sometimes, applying tExpand to a result from tCollect, or vice versa, in two separate steps simplifies an expression.

Some exercises:
Exercise 2.1. tExpand(cos(2 x))
Exercise 2.2. tExpand(sin(a+2b))
Exercise 2.3. tCollect(sin(x)sin(y))
Exercise 2.4. tCollect(cos(x)sin(y)tan(z))
Exercise 2.5. Find an example that shows tExpand and tCollect are not "inverse" to each other.

Section 3. trigonometric Identities, etc.

Let us play with tCollect and tExpand by using them to simplify some trigonometric expressions and establish some trigonometric identities.

It is important to know which of the two commands is more suitable for solving a given problem. Here is a simple way to remember the functions of tExpand and tCollect:
tExpand changes "multiple angles" into "single angles" while tCollect does the reverse.

Example 3.1. Simplify cos(x)^2 (1 + tan(x)^2) .

Type the following directly on the keyboard:
texpand(cos(x)^2(1+tan(x)^2)) ,
The answer is located on the lower right corner of the screen right above the input line---it is 1 in this example. (Try directly type the expression itself.)

Example 3.2. Simplify (tan(x)sin(x)-sec(x)) cos(x).

What should we do with sec(x)? As we mentioned earlier, TI-92 does not have a key for this function. A simple way to solve this problem is to use 1/cos(x) to replace sec(x). Another way is to define a function for sec(x). This is discussed in the next section. This second way is important if you are going to use sec(x) often when you use your TI-92. After this preparation, Ti-92 returns the right answer:
$-(cos(x))2$.
(Try directly type the expression itself.)

Example 3.3. Simplify sin(2 $cos-1(x)).$

As indicated in the exercise 1.6. in section 1, directly typing sin(2 $cos-1(x))\; will\; not\; do\; any\; thing\; on\; TI-92.\; But\; tExpand\; will\; help\; us:$
tExpand(sin(2 $cos-1(x)))$
will produce
2x$(1-x^2)^\{1/2\}$.

Example 3.4. Simplify $2sin(x)2-cos(x)2$.

Apply tExpand to the above expression will do the simplification. The answer is $3\; sin(x)2$-1.

Example 3.5. This example shows us a case in which TI-92 is helpless in solving some problems. (It is always possible to find a unsolvable problem for any given computer software.)

Let us first try to simplify
(1+cos(2x))/(2cos(x)).
Use tExpand, it will return the right answer: cos(x).

Next, let us simplify (1+cos(4x))/(2cos(2x)).
From the above, the answer should be cos(2x). But this time tExpand only returns something not as simple. Sometimes (maybe most times?), human interaction is important!

Some exercises:
Exercise 3.1. Simplify (1-cos(2x))/(2sin(x))
Exercise 3.2. Simplify (1-cos(6x))/(2sin(3x))
Exercise 3.3. Simplify tan(x)sin(x)-sec(x).

4. Solve trigonometric Equations

Example 4.1. Solve sin(x)=1/2.
Type solve(sin(x)=1/2,x) then . TI-92 will return
x=2 @n21 pi + 5 pi /6 or x=2 @n21 pi + pi /6
where @n21 denotes an arbitrary integer. (TI-92 has 256 such integers: @n1,@n2,...,@n255, @n0. It uses these integers in this order. When @n0 is used it will be repeatedly used in the subsequent calculation when an arbitrary integer is needed.) Example 4.2. Solve sin(x)=cos(x) /2.
Directly use solve(sin(x)=cos(x)/2,x) is slow (about 45 sec.) on TI-92 and the answer is not in the general form. Try solve(tan(x)=1/2,x) followed by . This will speed up TI-92 and the solution is in the general form.

5. How to Define Functions in TI-92

There are three ways to define a function.

1. Use the Define command.
Define sec(xx)=1/cos(xx) followed by
will define sec( ). It will stay in the memory.

2. Use the store-key .
1/sin(x) csc(xx)
will create a function csc( ).

3. Write a simple program. Do the following to create a function cot( ):
, 7, choose 2 in Type, , move the cursor down to Variable and then type cot, , .
A screen like this

:cot( )
:Func
:
:
:EndFunc

will appear. Modify it so that it looks like this:

:cot(xx)
:Func
:cos(xx)/sin(xx)
:
:EndFunc

Now, we can return to Home (, ). If you type cot(x) sin(x) , , Ti-92 will return cos(x).

6. Some Useful Tips

I plan to update this list as soon as I learn some new tricks/shortcuts. Please kindly send me yours.

1. Use then cursor to move the cursor to the end (and beginning) of line. In Data\Table Editor, this can help us to move the cursor to the top (or bottom) of a column.

2. There are two keys: one on the right-hand side, and another, the left-hand side.

3. F1 8 clears the Home memory.

4. followed by return us to the Home page. Another way to go "Home":

Use the shortcut key followed by Q key.

5. 5 (math)
lists all topics in mathematics available in TI-92.

6. 2(catalog)
provides an alphabet list of everything in TI-92.

7. -(var-link)
allows us manage variables we create on the way.

8. ans(1) is the last answer, ans(2) is the second to the last answer, ans(3) ...

9. Use cursor to highlight entries in the history area of the screen then use to copy the highlighted entry to the entry line of the screen.

Answer to the question near the end of section 1:

The answer returned by TI-92 is still $sin-1(sin(x))$. The reason is that the answer really depends on what the range of the value of x is, and TI-92 is smart enough to wait for that information to be given to decide its output. How can we tell TI-92 the range of x, say [-pi/2, pi/2]?
Ti-92 has a "with" key: | . It is located as the second function on . For example, if you type x+1 | x=2 , , then the answer is 3. This with key is very powerful. See its definition on page 468 in the user's guidebook. Now back to our question, we need to type the following:
$sin-1(sin(x))$ | x>=-pi/2 and x<=pi/2
This will give us x.

Please send your comments to me at xli@pegasus.cc.ucf.edu. (Updated 8/24/96)