Part 3 - TI-89/Voyage 200/TI-92+facstaff.cbu.edu/wschrein/media/M231 Notes/TI89manual.pdf · Part 3 - TI-89/Voyage 200/TI-92+ Chapter 1 – Basics A. About the TI-89, Voyage 200,

Part 3 - TI-89/Voyage 200/TI-92+

Chapter 1 – Basics

A. About the TI-89, Voyage 200, TI-92+, TI-92All of these calculators have symbolic algebraic capability. This means that they can solve equations and docalculus using letters. Therefore, they may not be allowed in certain classes or on certain exams. The TI-89 isabout the same size as the TI-83+, and may be more acceptable because it does not have a QWERTY keyboardlike the TI-92+ and Voyage 200. The TI-89, 92+ and Voyage 200 can run flash applications and have theiroperating systems upgraded. These can be downloaded from http://education.ti.com/us/product/prselect.html.The TI-89 and Voyage 200 are powerful, but it takes about twice as many button presses to do the same task asit does on a TI-83+. Hence, some students with TI-89s have trouble following an instructor who is using a TI-83+. The TI-89 and above also have unit conversions and physical constants built in.

The TI-92 cannot run flash applications nor have its operating system upgraded like the TI-89 and Voyage 200.However, most of the instructions given in this section of the manual will also work for the TI-92.

B. General CommandsIn addition to the obvious purpose of the ON key for turning on the calculator, it will also halt the execution ofa program and stop the progress of graphing.

For a quick reference on the syntax of a command, you can use the alphabetical listing of operations at the endof the manual that came with your graphing calculator. The screen that you see when first turn on the calculatoris the home screen; this is the screen where most calculations are done. All functions in yellow on the face ofthe calculator are obtained by pressing the 2nd key and the desired key. The characters in purple are obtained bypressing the alpha key and the desired key. All functions in green on the face of the calculator are obtained bypressing the green diamond key and the desired key.

If you cannot remember the menu in which to find a command, you can always find an alphabetical list in thecatalog by pressing CATALOG. For example, if you are looking for the rref( command, you can do R toquickly move to the commands starting with the letter R. Then use down arrow key to select rref( and then pressENTER.

C. The MODE keyPress the MODE key and use the arrow and ENTER keys so that the settings are as shown in the picture. PressF2 or F3 to see the other two pages of modes. These are the usual settings, but there will be a few that you maywant to change later. Press the ENTER key to save the changes you made and to return to the home screen.

Page 1 Page 2 Page 3

D. Entering ExpressionsAlgebraic expression evaluation obeys the “Please Excuse My Dear AuntSally” rule. In decreasing order of priority, the operations are

1) Parentheses

Sally” rule. In decreasing order of priority, the operations are 2) Exponentiation3) Multiplication and Division4) Addition and Subtraction

See your calculator manual for a more specific description of the order of evaluation. Note that there areseparate buttons for subtraction – and negation (-). If you are ever in doubt as to the how your calculator willevaluate an expression, then use parentheses.

The ENTRY and ANS keys are useful for doing complicated calculations. The ENTRY key retrieves aprevious expression, while the ANS key retrieves a previous numerical result. Alternatively, you can up arrowinto the history screen and press ENTER to bring a desired expression or value to the command line. Note thatan expression can be edited using the delete DEL, insert INS, and arrow keys. Pressing ENTER will cause anexpression to be evaluated or command to be executed.

Scientific notation is expressed using the EE key. Note that although thiskey shows two Es, only one is displayed on the screen. When you writeresults on paper, you should use standard notation, not calculator notation.For example, the number on the command line shown here should bewritten as 12103 −× on paper.

Back to the Table of Contents

Chapter 2 – Graphs and Technology

A. Basic GraphingPress the Y= key and enter the rightside of the equation xxy −= 3 aftery1. Select F1 (Tools)\9:Format andmake sure the settings are as shownhere. These are the settings that will beappropriate for most graphing. PressENTER to save your settings.

Select F2 (Zoom)\ZoomDecimal and agraph should appear. Press WINDOWto see the x and y ranges correspondingto zoom decimal and change xres to 1or 2. You can press GRAPH to see thegraph again.

Your calculator graphs by plotting points and connecting the points with line segments. If you look carefully atthe calculator screen when your calculator is turned on, you might see a grid of tiny rectangles. Each rectangleis either turned on (made dark) or turned off (made light) to make a graph. xres is the number of pixels that yourcalculator skips in the horizontal direction when plotting the endpoints of the line segments. xres can range from1 to 10. A small value of xres will give a smoother graph, but it takes longer to complete the graph.

Exercise: Redo the above example using an xres of 8. Explain the shape of the graph.

B. A Good GraphA good graph should definitely clearly show the structure that you are interested in. For example, if you aresolving for where an expression is zero, the corresponding x intercept should be clearly visible. It is usuallypossible and preferable to choose a window so that the graph clearly shows all the important features like x-intercepts, y-intercepts, maximums, minimums, asymptotes, and behavior as x becomes large. The zoomdecimal window chosen in Example A was a good choice for graphing the equation xxy −= 3 .

Exercise: Try to find a good window that will show all the important features of xxxy 2021 23 +−= . Then,find all the x-intercepts by factoring. Did your “good” graph clearly show all the x-intercepts?

C. Missing Parts of GraphsSometimes your calculator leaves gapsin a graph. This can happen when anequation is defined on one end andundefined at the other of a line segmentthat your calculator is trying to plot. For

example, graph 26 xy −= in thewindow shown here.

The graph should be a completesemicircle, but your calculator leavesoff the ends. When you copy a graph topaper, you should fill in details thatyour calculator omits.

Exercise: Do zoom square so that thesemicircle is not stretched out.

Graph given by your calculator Graph as it should appear

D. Graphing AsymptotesSometimes your calculator will graph vertical lines that are not part of the graph. This can happen when a graphblows up to positive infinity on one side of the line and negative infinity on the other side.Such lines are called verticalasymptotes, and since they are not partof the graph, they should be graphed asdashed lines. For example, graph the

equation 32

2

−=x

y in the window

shown here.

See that your calculator graphs the

vertical line 2

3=x . This line should be

drawn as dashed when transferring thegraph to paper.

Graph as drawn by calculator Graph as it should appear

Exercise: Redo the example in Example D using a zoom decimal window. (Even though your calculator doesnot graph the vertical line, you should still graph it as a dashed line on paper.)

E. Showing an Equation Is an IdentityExercise 41 in Section 2.1 of the textbook explains how to verify if an equation is an identity or not by graphingboth sides of the equation. Instead of graphing the sides separately and tracing, you can graph only 121 +− yy .The equation is an identity if and only if the graph of 121 +− yy is the horizontal line 1=y . (Can you explainwhy?) Before you graph, deselect y1and y2. The y1 and y2 symbols used for y3 are simply typed using the ykey and the numbers 1 and 2.

Here is the technique applied to showthat 2510)5( 22 ++=+ xxx is anidentity. A zoom decimal window wasused. Since the graph is the horizontalline 1=y , we are fairly confident thatthis equation is an identity.

F. Showing an Equation Is Not an IdentityHere is an example using the techniquegiven in Example E showing that

25)5( 22 +=+ xx is not an identity. Azoom decimal window was used. Sincethe graph is a not horizontal line, weare certain that this equation is not anidentity.

G. Linear RegressionRefer to Example 2 in Section 2.5 of the textbook. Although there is a data editor built into the TI-89 andVoyage200, the 83+ style statistics/list editor application is easier to use and more powerful. Also, it is easier tofollow your teacher if he or she is using a TI-83+. This application can be downloaded from TI’s websitehttp://education.ti.com/us/product/prselect.html. After you have installed this application, run it by selectinggreen diamond APP\Stats/List Editor\ENTER. We will use L1 and L2 for the names for the x and y lists,respectively. Note that the TI-89 and Voyage 200 do not distinguish upper and lowercase letters. Clear out allfunctions under the Y= key. To set up a scatter diagram, you can press ENTER over the Plot 1 symbol in theY= menu. Then select the scatter diagram icon, and select lists L1 and L2 as shown in the picture; pressENTER to save the plot definition. To quickly find a window for the scatter diagram, do ZOOM\ZoomData.Press Trace and the left and right arrow keys.

To find the least-squares regressionline, go back to the statistics/list editorapplication. SelectF4 (Calc)\Regressions\LinReg(ax+b).Then fill in the dialog box so thecalculator knows you are using lists L1and L2, and you want the regressionequation stored in y1(x).

You should write the regressionequation on paper as 4.13423. += xy ;do not just write the values of thecoefficients a and b. Press Y= to verifythat the regression equation is in y1.Then press GRAPH to see theregression equation graphed with thescatter diagram.

Suppose you want to predict the poverty level in 2005. Since the regressionequation is already in y1, go to the home screen and evaluate this functionat 15. (The TI-89 and Voyage use function notation that will be explainedin Chapter 3, Section A.) Thus, the predicted poverty level is $19,753.

Prediction using the home screen

Exercise: Predict the poverty level in 2005 graphically by doingGRAPH]ZoomOut\ENTER and then F5 (Math)\Value\15\ENTER.


Prediction using the graph screen

Chapter 3 – Functions and Graphs

A. Evaluating a Function

We will evaluate 1)( 2 += xxf usingfunction notation for the x value of 3and the x value of 5- . First, put thisfunction in y1, and then go to the homescreen. Note that including a decimalpoint in the expression caused thecalculator to give approximate valuesfor 10)3( =f and 26)5( =−f . Thiscould also have been accomplished bypressing green diamond ENTER.

pressing green diamond ENTER.

B. Greatest Integer Function [x]Put this function in y1 by pressing MATH\Number\floor(. To make a table of values for this function, pressTblSet and set the parameters as shown in the screen below. Press TABLE and try scrolling through the tableusing the up and down arrow keys.

Graph the greatest integer function in a zoom decimal window. If your calculator is in connected mode, then itwill graph vertical parts of the steps that are really not part of the graph. Go to the Y= screen, pressF6(Style)\Dot, and then press GRAPH again to obtain a more correct looking graph. Try tracing your graph tosee the x values where it jumps. The actual graph shown at the end is how you should graph ][x on paper.

Graph of ][x in connected mode Graph of ][x in dot mode Actual graph of ][x

C. Piecewise-defined Function

The piecewise-defined function

≤<+

≤=

41if2

1if)(

2

xx

xxxf was graphed in Example 6 of Section 3.3 of the

textbook. To graph it on your calculator, type the function in y1 as undef))2,x4,when(x,x1,when(x 2 +≤≤ .The ≤ symbol can be obtained by typing green diamond <, “when” can be found in the CATALOG or typedusing the alpha key, and “undef” must be typed using the alpha key. Use the window values shown below andgraph in dot mode. Note that )(xf is not defined if 4>x .

D. Minimum of a Function

This example refers to Example 8 in Section 3.3. Enter the function 18.1)( 23 ++−= xxxxf in y1 and try tofind a window that clearly shows the extrema without looking at the graph range shown here. Tracing will onlygive points that are within a pixel of the true minimum. To find and accurate value for the minimum, pressGRAPH\F5(MATH)\Minimum and answer the prompts using the left and right arrow keys and pressingENTER. The minimum value for the function f is 1.1593=y and it occurs at 7633.=x .

E. Parametric GraphingThis example refers to Section 3.3 Example 13. Press MODE\Graph\PARAMETRIC\ENTER to put yourcalculator in parametric mode and then enter the functions 12 −−= ttx and )32(643 ≤≤−−−= ttty underthe Y= menu. Press WINDOW and enter the values shown below. You can easily find a good window by doingF2(Zoom)\ZoomFit. F3(Trace) the graph to see how the ),( yx points correspond to the t parameter values.

Exercise: Try differenttStep values in theprevious example. Whathappens if tStep is toosmall? What happens iftStep is too large?

You can also make a table showing thepoints that your calculator is plotting tocreate the parametric graph. Adjust thevalues under TblSet as shown in thefirst screen, and press TABLE to seethe second screen.

F. Power Functions xn

This example explores the shape of the power functions by putting a list in a function.

Enter the functions as shown in the firstscreen, and deselect y2. Then graph in azoom decimal window. This will graph3x , 5x , and 7x simultaneously. Notice

that all these graphs are symmetric withrespect to the origin. In fact, all powerfunctions with odd exponents are oddfunctions.

Now select y1 and deselect y2. Thengraph in a zoom decimal window. Thiswill graph 2x , 4x , and 6xsimultaneously. Notice that all thesegraphs are symmetric with respect tothe y-axis. In fact, all power functionswith even exponents are even functions.

G. Vertical Shift

Put the square root function in y1 andselect a thick line plotting style. Add 4to the outside of the function y1 to shiftit up 4 units. Use a zoom standardwindow to obtain the graph shownhere.

Exercise: Trace y2 in order to estimate its domain and range.

A table can also be constructed todemonstrate a vertical shift. PressMODE\Complex Format\REAL\ENTER

so that x will be undefined fornegative values instead of returningnonreal values. Note that all the y valuesof y2 in the table are 4 greater than the yvalues of y1

H. Horizontal Shift

Put the square root function in y1 andselect a thick line plotting style. Add 2to the inside of y1 to shift it to the left 2units. Use a zoom standard window toobtain the graph shown here.


A table can also be constructed todemonstrate a horizontal shift. Notethat all the y values in the y1 columnare shifted 2 units up to get the y valuesof y2 column.

I. Vertical Expansion

Put the square root function in y1 andselect a thick line plotting style.Multiply the outside of y1 by 2 toexpand it vertically by 2. Use a zoomstandard window to obtain the graphshown here.


A table can also be constructed todemonstrate a vertical expansion. Notethat all the y values in the y2 columnare twice as large as the y values in they1 column.

J. Vertical Contraction

Put the square root function in y1 andselect a thick line plotting style.Multiply the outside of y1 by 0.5 tocontract it vertically by 0.5. Use a zoomstandard window to obtain the graphshown here.


A table can also be constructed todemonstrate a vertical contraction. Notethat all the y values in the y2 columnare half as large as the y values in they1 column.

K. x-axis Reflection

Put the square root function in y1 andselect a thick line plotting style. Negatethe outside of y1 to reflect it about thex-axis. Use a zoom standard window toobtain the graph shown here.


A table can also be constructed todemonstrate an x-axis reflection. Notethat all the y values in the y2 columnnegatives of the y values in the y1column.

L. y-axis Reflection

Put the square root function in y1 andselect a thick line plotting style. Negatethe inside of y1 to reflect it about the x-axis. Use a zoom standard window toobtain the graph shown here.


A table can also be constructed todemonstrate a y-axis reflection. Notehow the y values in the y2 column arerelated to the y values in the y1 column.

M. CompositionThis example refers to Example 5 inSection 3.5. Put xxf =)( in y1 and

5)( 2 −= ttg in y2 and deselect both ofthese functions. The composition fg ois shown in y3. Graph y3 in the windowshown below. Note that the graphagrees with the rule for the composition( ) 0,5)( ≥+= xxxfg o that wasderived in the textbook.

Exercise: Find the domain and range of the function fg o .

N. Instantaneous Rate of Change

Refer to Example 9 of Section 3.6 of the textbook; the volume of a balloon is given by 55

)(3x

xV = in gallons

where x is the radius of the balloon in inches. A table can be used to find the instantaneous rate of change of Vwhen 7=x inches. Put the formula for the volume of the balloon in y1 and the difference quotient

h

VhV )7()7( −+ in y2. Set the independent variable to Ask under TblSet and enter decreasing values of h for x

under TABLE as shown below. As x converges to zero, the difference quotient converges to 2.673. Thus, therate of change of the volume at the instant 7=x inches is 2.673 gallons/inch.

O. Inverse Function

Put the function 5)( 3 += xxf in y1 using a thick graphing style, the function 3 5)( −= xxg in y2, and theidentity function in y3. The first graph was done using a zoom standard window. Zoom square was appliedagain to obtain the second graph so that the line xy = would appear diagonal. Note that the functions f and gare reflections of each other about this line because they are inverse functions.

If you are starting with only5)( 3 += xxf , then you can easily

graph its inverse by executing theGRAPH\F6(Draw)\DrawInv y1(x)command on the home screen. Aninverse function cannot be traced whengraphed using the draw inversecommand.

Another way to graph both a function and its inverse is to put your calculator in parametric mode. The firstparametric function contains f, the second contains g, and the third contains the identity function. First do zoomstandard in order to adjust the x and y window ranges, and then use the t range shown here. Lastly, do zoomsquare.


Chapter 4 – Polynomial and Rational Functions

A. Evaluating PolynomialsThe usual way of writing a polynomial is to write it as a sum of monomials with decreasing powers. Forexample, the polynomial 87725)( 234 ++++= xxxxxf is written in the standard form. Your calculator will

evaluate a polynomial faster if it is written in nested form. For example, ( )( )( ) 87725)( ++++= xxxxxf is thesame polynomial written in nested form. After some practice, you can quickly write down the nested form of apolynomial. The derivation of the nested form involves successively factoring out powers of x. Here is aderivation for this polynomial:

( )( )( )( )( )( )( ) 87725

87725

87725

87725)(

2

23

23

++++=

++++=

++++=

++++=

xxxx

xxxx

xxxx

xxxxxfThe reason your calculator will evaluate a polynomial more quickly innested form is multiplication is performed faster than exponentiation. Forthe polynomial given here, the standard form has 3 exponentiations and 4multiplications; the nested form has only 4 multiplications.

Exercise: Test that the nested form of a polynomial is evaluated quicker on your calculator than the standardform. Put the standard form of the polynomial )(xf given in the above example in y1 and the nested form iny2, and deselect y2. Set xres=1, find a good window, and then time how long it takes your calculator to graphthe standard form. Next, deselect y1, select the nested form in y2, and time how long it takes the nested form tograph the nested form. How much faster did the nested form graph?

Select only the standard form Select only the nested form A good graph of f

There is a special command on the TI-89 and Voyage 200 for evaluating polynomials. The polynomial )(xfwould look like )},,8,7,7,2,5({polyEval x on your calculator. The polynomial evaluator command is under theCATALOG, and the braces are above the parentheses keys. The coefficients for the polynomial appear in thebraces separated by commas, so do not forget to put zeros in for any missing coefficients. The last argument ofthe polyEval is where you want to evaluate the polynomial.

Exercise: Test if the polyEval Command graphs faster than the nested form.

B. Polynomial DivisionThe TI-89 and Voyage 200 can symbolically divide polynomials. Forexample, suppose you want to divide 87725)( 234 ++++= xxxxxf by

5)( 2 ++= xxxg . First do F6\(Clean Up)\Clear A-Z in order to delete thecalculator variable x. Then perform the following operations on the homescreen:

fxxxx →++++ 872^73^24^5gxx →++ 52^

calculator variable x. Then perform the following operations on the homescreen:

This will store the dividend and divisor in the calculator variables f and g.(The → is the STO key.) Lastly, perform the proper fraction command asshown on the screen. This command is under the F2 (Algebra) menu. Thus,the quotient is 1535 2 −− xx and the remainder is 8337 +x . Note that youcould also write the relationship between these polynomials as

( ) ( )8337)(1535)( 2 ++−−= xxgxxxf .

C. Polynomial SolverAlthough the TI-89 and Voyage200 do not have a polynomial solver built in, but a free flash application thatfinds the approximate zeros of polynomials can be downloaded from TI’s websitehttp://education.ti.com/us/product/prselect.html. Suppose you want to solve the equation 018124 35 =−+− xxxgiven in Example 5 of Section 2.2 in the textbook. After you have installed this application, run it by selectinggreen diamond APP\Polynomial Root Finder\ENTER\New… The application first prompts for the degree ofthe polynomial. After typing in 5, press ENTER to go to the next screen where the coefficients of thepolynomial are entered. Note zeros are entered for the missing coefficients. Press F5 (Solve) to find theapproximate roots. Note that this polynomial equation has 5 distinct real roots.

D. Quick Bounds for Zeros and Graphing PolynomialsThere is a faster, although less accurate test for bounding the zeros of a polynomial than the Bounds Testintroduced in Section 4.3 of the textbook. In general, if z is a zero of the polynomial

011

1)( axaxaxaxf nn

nn ++++= −

− L where 0≠na , then ( )

1||

|||,||,||,|max|| 0121 +≤ −−

n

nn

a

aaaaz

L. For

example, consider the polynomial 20363328796)( 234567 +−+−++−= xxxxxxxxf . The largest the

absolute value of a zero for f can be is ( )

3711

20,36,33,28,7,9,6max=+ .

Thus, every real zero of f is guaranteedto be in the interval 3737 ≤≤− x . Thiscan be used to start looking for a goodgraph that will not omit any xintercepts. Enter this polynomial in y1.

(You do not have to use the polyEvalcommand.) Use the graph windowshown here.

This first graph shows that all thereal zeros are in the interval

55 ≤≤− x . Use this for the xrange and do zoom fit. Adjust thewindow again to get a good graph.

Exercise: Do zoom box in order to see the positive x-intercepts more clearly.

E. Horizontal Asymptotes

Define the rational function 2

2)(

2

2

−+=

xx

xxf . In Example 5 in Section 4.3 of the textbook, it was shown that

2=y is the horizontal asymptote for f. See the screens below to see how to graph this rational function with itshorizontal asymptote. (The function y1 has a thick-line graphing style.)

You can also see that 2=y is a horizontal asymptote by making a table. Set up the table so that it asks for theindependent variable. Then you can see that as x approaches either infinity or negative infinity that )(xfapproaches 2.

Exercise: Why is y1 undefined when when 1=x in the above table?

F. Oblique Asymptotes

In Example 2 of Section 4.5.A, it was shown that 432 −+= xxy is the oblique asymptote for the function

1

572)(

23

−

+−+=

x

xxxxg . The screens below show how to graph g with its oblique asymptote. (Give y1 a thick-

line plotting style.)

This can also be demonstrated with a table. Deselect y1 and y2 and put the difference in y3. Set up the table sothat it asks for the independent variable. Then you can see that as x approaches either infinity or negativeinfinity, that the difference between )(xf and its oblique asymptote approaches zero.

Exercise: Why is there an error when 1=x in the above table?

G. Complex Arithmetic

Press the MODE key and select Complex Format\RECTANGULAR sothat your calculator will do complex numbers and do them in standard form.Your calculator will do addition, subtraction, multiplication, division, andexponentiation of complex numbers. They are typed on the home screenjust like they appear in the textbook. The complex constant i is above theCATALOG key on your calculator.

This example shows how to multiply two complex numbers, raise i to apower, and divide two complex numbers.


Chapter 5 – Exponential and Logarithmic Functions

A. Root Command

The identity nn xx /1= must be used to compute a root. The screen shownto the right demonstrates taking the 4th root of 81.

Exercise: Graph the functions xy = , 3 xy = , 4 xy = , and 5 xy = in azoom decimal window. How does the shape of the graph, the domain, andthe range of n xy = depend on n?

B. Exponential and Logarithmic Functions

In this example, we will graph the exponential function xy 10= , and its inverse function, the common

logarithm xy 10log= in a zoom decimal window. The commands for these functions can be found either under

the CATALOG or can be simply typed using the alpha key. Since they are inverse functions, they arereflections of each other about the diagonal line xy = . Trace y1 to see that xy 10= does not touch the x-axis.

Also, xy 10log= does not touch the y-axis.

Exercise: What is the domain and rangeof xy 10= ? What is the domain and

range of xy 10log= ? Do your answers

agree with the graph?

Exercise: Graph xey = and xy ln= with xy = . Then write down the domains and ranges for xey = andxy ln= and relate them to your graph.

C. Nonlinear Regression

Regression curves can also be found for the models studiedin Chapter 5. The first column in the table shown at theright gives the names of the models, the second columngives the formulas for the curves, and the third columngives the calculator commands for finding those regressioncurves. These commands are under the STAT\CALC menuand the syntax is the same as for linear regression.

model formulacalculatorcommand

logarithmic xbay ln+= LnReg

exponential xaby = ExpReg

power baxy = PowerReg

logistic likeon the TI-83 bxae

cy

−+=1

Logistic83

logistic likeon the TI-86

dbe

ay

cx+

+=

−1Logistic

Refer to Example 1 in Section 5.6 ofthe textbook. Enter the data in the listsL1 and L2 using the statistics editorflash application. Then set up a scatterdiagram and do zoom data. Notice thatthe shape of the scatter diagramsuggests that an exponential model isappropriate.

Select F4(Calc)\Regressions\ExpReg on the home screen to find that the best-fitting exponential regressioncurve is ( )xy 0299.19572.3= .


Chapter 6 – Trigonometric Functions

A. Degrees and RadiansA common mistake when evaluating trigonometric functions is to be in the wrong angle mode. An angle modecan be set by pressing the MODE key and then selecting DEGREE or RADIAN after Angle. Press ENTER tosave your choice and return to the home screen. Note that your calculator displays the angle mode at the bottomof the home screen.

Setting the mode to degrees Setting the mode to radiansThe angle mode is indicated by DEG orRAD at the bottom of the screen.

B. Cosecant, Secant, CotangentThese functions come with operating system version 2.08 for the TI-89 andVoyage 200, but there is a bug that TI has not fixed yet involving them.Therefore, it is recommend that you use the reciprocal identities

ttsin

1csc = ,

ttcos

1sec = and

tttan

1cot = with the SIN, COS, and TAN

keys, respectively, instead. For example, if you wanted to approximate°10cot , then put your calculator in degree mode, and execute the statement

as shown on the home screen as shown here.

C. Evaluating Trigonometric Functions

Use parentheses when in doubt as to how your calculator will evaluate anexpression. For example, the next screen shows how to evaluate x2sin at

radians9.1=x . Do not forget to put your calculator in radian mode first!

The first screen shows two correct waysto evaluate x2sin at radians9.1=x .(Your calculator does not understandthe power notation x2sin , so you willhave to use the definition

22 )(sinsin xx = .) The second screen

shows a wrong way to compute x2sin . Correct value of x2sin This is ( )2sin x , not x2sin .

D. Zoom Trig Window for the TI-89This is a convenient window to use when graphing trigonometric functions. Although graphing is usually donein radian mode, keep in mind that the x range depends on the angle mode. After selecting radian or degree

mode, press ZOOM\ZoomTrig. If your calculator is in radian mode, then the x range is approximately π4

13−

to π4

13 with an x scale of 2/π . If your calculator is in degree mode, then the x range is approximately °− 590

to °590 with an x scale of °90 .

Radian zoom trig window Degree zoom trig window ty sin= graphed using zoom trig

Exercise: Approximately how may periods of ty sin= appear in the zoom trig window shown above?

D. Zoom Trig Window for the Voyage 200This is a convenient window to use when graphing trigonometric functions. Although graphing is usually donein radian mode, keep in mind that the x range depends on the angle mode. After selecting radian or degreemode, press ZOOM\ZoomTrig. If your calculator is in radian mode, then the x range is π5− to π5 with an xscale of 2/π . If your calculator is in degree mode, then the x range is approximately °− 900 to °900 with an xscale of °90 .

Radian zoom trig window Degree zoom trig window ty sin= graphed using zoom trig

Exercise: Approximately how may periods of ty sin= appear in the zoom trig window shown above?

E. Amplitude, Period and Phase ShiftWe will redo Example 1 in Section6.5.A of the textbook in more detail. Inthat example, the constants cbA ,, wereestimated where

)43cos(2)23sin(4

)sin(

−++=

+

tt

cbtA.

Graph the right side of this equationusing the window shown here.

Use your calculator’s maximumcommand to find that the amplitude is3.94. Press HOME and store the xcoordinate of this maximum in thecalculator variable A.

Find the next maximum and return to the home screen. Then subtract the two x values to find that the period is

2.09. Solve b

π2 period = for b to get

period

2π=b . Ignoring round-off error, the calculator agrees the correct

value of 3=b .

Finding the next maximum Computing the period Computing b

We will first compute the phase shift and then find the constant c. Put the function )3sin(94.3 ty = in y2 andpress GRAPH. y2 can be shifted either left or right to obtain y1, but we will assume the phase shift is negativebecause this magnitude of shift is less. Find the maximum of y2 as shown and subtract its location from thelocation of the original maximum found for y1 to find that the phase shift is -0.836. Note that the phase shift isnegative because we decided that y1 should be shifted left to obtain y2.

Solve b

c−=shift phase for c to get ( )shift phasebc −= . Use this last

equation to find that 51.2=c . Thus,)51.23sin(94.3)43cos(2)23sin(4 +=−++ ttt .

F. Sine RegressionRefer to Example 11 in Section 6.5 of the textbook. Enter the month data in list L1 and the temperature data inlist L2 in the statistics editor flash application.

Do not forget to duplicate the data sothat the months run from 1 to 24. Thencreate a scatter diagram by doing zoomdata. Sine regression is appropriatebecause the shape is a sine wave.

In the flash statistics editor, sine regression command is under the F4\CALC\Regressions\SinReg menu. Fill inthe dialog box as shown below and press ENTER.

Press the right arrow key to see the rest of the regression coefficients. Youshould write the regression equation as 6.49)18.2522.0sin(7.22 +−= tyon paper. Press the GRAPH key to see how the regression equation fits thescatter diagram.

Exercise: Why is the period in the above example equal to 12?


Chapter 7 – Trigonometric Identities and Equations

A. Showing an Equation is Not an IdentityRefer to Example 1(a) of Section 7.1 in the textbook. One way to check if an equation is an identity is to graphboth sides of the equation. For example, to determine if the equation xxxx sincos2cossin2 22 +=− is anidentity, put the left side in y1 and the right side in y2. Note that the plot style for the right side is a thick line.Recall that all graphing is done in Radian mode. A ZoomTrig window was used for the graph shown here. Sincethe graphs of both sides of the equation are not the same, the equation is not an identity. To show an equation isnot an identity, it is also sufficient to find a single value for which both sides are different. The table feature canbe used to also see that both sides are not equal.

B. Showing an Equation is an IdentityRefer to Example 1(a) of Section 7.1 in the textbook. We will use the technique demonstrated in Section A to

show that xxx

xxtancos

cos

sinsin1 2

+=−+

is an identity. Note that both graphs are exactly on top of each other

and the two functions agree for all the x’s in the table. You can trace the graphs to make it easier to see that thefunctions are the same. Note that we have not proven that this equation is true for all x. We only have strongstatistical evidence because both sides agree for all the x’s that your calculator graphed.

Instead of graphing the functions separately and tracing to make it easier to see that both sides are the same, youneed only graph 121 +− yy . The equation is an identity if and only if the graph of 121 +− yy is the horizontalline 1=y . (Can you explain why?) Before you graph, deselect y1 and y2.

C. Checking an Exact Answer

In Example 5(a) of the textbook, it was shown that the exact value of

8

5cos

π is

2

22 −−. To check this, ensure that your calculator is in

AUTO mode. Note that a large number of decimal places does not mean thenumber is exact.

D. Graphing an Inverse Trigonometric Function

A graph of xsin in a ZoomTrigwindow shows that the sine function isnot one-to-one.

You can graph the sine function restricted to 22

ππ≤≤− x by putting

)undef),sin(,2/and2/(when xxx ππ ≤−≥ in y1. The “when” command is under the CATALOG; theinequalities can be obtained by doing green diamond > and green diamond >; the “and” symbol is under theMATH\BASE menu; just type the “undef” symbol using the alpha key. Use ZoomBox to get a larger graph ofthe restricted sine.

Note that this function is one-to-one. Select the thick line style for y1 and put the identity function in y2. Tograph the inverse of the restricted sine, execute the draw inverse command as shown in the middle window.This command is found under GRAPH\Draw\DrawInv. The third window shows the graph of sine and inversesine reflected across the line xy = . Note that you cannot trace an inverse function when it is drawn in thismanner.

Exercise: Redraw the inverse sine function using ZSQR.Exercise: Draw the inverse sine function using the SIN-1 button on your calculator.

E. Composing a Trigonometric and Inverse Trigonometric FunctionIn section 7.4 of the textbook, it isshown that uu =− sinsin 1 if

22

ππ≤≤− u . This is demonstrated

using the technique shown inSection A. Note that uu =− sinsin 1 byitself is not an identity.

In section 7.4 of the textbook, it isshown that vv =−1sinsin if

11 ≤≤− v . This is demonstrated usingthe technique shown in Section A. Notethat vv =−1sinsin is an identity.

F. All the Solutions of a Trigonometric EquationRefer to Example 1 in Section 7.5 of the textbook where all solutions to 2tan =x were found. Both sides of theequation were graphed using an x range of ππ 42 ≤≤− x and a y range of 42 ≤≤− y . Note that theintersection of the vertical asymptotes and the line 2=y do not give solutions to the equation 2tan =x . Usethe Intersection command under the GRAPH\Math menu to find the smallest positive intersection of xy tan=and 2=y .

Since the tangent function has period !, all the relevant roots are separated by !. Thus, the solution set is givenby πkx += 1071.1 where k is an arbitrary integer. You can verify that this formula gives solutions to theequation 2tan =x by using the table feature of your calculator:

Use the solve( command that is found under the F2 (Algebra) menu todirectly find all the roots of the equation 2tan =x . First, ENTER waspressed giving the exact solution in terms of the inverse tangent function.Then, green diamond ≈ was pressed to give the approximate solution thatwe had found graphically. The @n2 symbol represents an arbitrary integer.

G. Checking an Identity in Two VariablesWe will graphically show that )sin()sin()cos()cos()cos( yxyxyx +=− is an identity using a technique similarto the one shown in Section B. Since this equation involves two variables (x and y), our graphs will involvethree variables (x, y, and z) and will be surfaces in space.

Put your TI-89 in 3-dimensionalgraphing mode by doingMODE\Graph\3D. Press Y= to see the3-D function editor. Put the left side inz1 and the right side in z2, and thedifference in z3. Deselect z1 and z2.Then press F1\Format and ensure thatthe settings are the same as shown here.

Press WINDOW and set the x and yranges from π2− to π2 ; set the zrange from -2 to 2. Press GRAPH tosee the graph of z1-z2 as a function of

),( yx . Since the graph is the plane0=z , this confirms that 21 zz = . Try

tracing and pressing the arrow keys tomove around the grid and see that allthe z’s are approximately zero.


Chapter 8 – Triangle Trigonometry

A. Checking a Law or Cosines or Sines Problem

This example demonstrates how to check that you have found all the remaining parts ofan oblique triangle correctly. Refer to Example 1 in Section 8.3 of the textbook. Aftersolving the triangle, all the parts were found as shown here. °==

°==

°==

1106.21

8.2510

2.4416

Cc

Bb

Aa

Every triangle satisfies Mollweide’s Equation: 2

sin2

cos)(BA

cC

ba−

=

−

If your values for the sides and angles satisfy this equation, then it is likelythat you have not made an error. Likewise, if your values for the sides andangles do not satisfy this equation, then it is likely that you have made anerror. First, put your calculator in degree mode. Then evaluate the bothsides of Mollweide’s Equation as shown in the calculator screen. Both sidesagree to roughly three significant digits, confirming that the sides andangles are correct.

Exercise: Why do both sides of Mollweide’s Equation only agree to three significant digits in the aboveexample?

B. Solving an Oblique TriangleThe triangle having the three parts 5.7=a , 12=b and °= 35A was solved in Section 8.3, Example 4 of thetextbook. In this example, it will be shown how to solve this triangle by doing a chain calculation on the homescreen. This is an example of the SSA case where there are two possible triangles that satisfy the given

information. Solve the Law of Sines formula 5.7

35sin

12

sin °=

B to find 92.0sin =B .

(Although only 3 significant digits are being written down in thisparagraph, all the digits are kept in the calculator to avoid round-off error.)Since there will be two triangles, the parts of the first triangle will besubscripted with a 1, and the parts of the second triangle will be subscriptedwith a 2. Apply the inverse sine to find the angle °= 6.661B . This is storedin the calculator variable B because it will be needed for the secondtriangle. Since the sum of the angles in a triangle is °180 , find

°=−−°= 4.78180 11 BAC . Solve 5.7

35sin4.78sin

1

°=

°

c to find

8.12sin

sin 11 ==

A

Cac . It remains to solve the second triangle. Take the

supplement to find °=−°= 113180 12 BB . Since the sum of the angles in a

triangle is °180 , find °=−−°= 6.31180 22 BAC . Redo the previous Law

of Sines formula to get 85.62 =c .


Chapter 9 – Applications of Trigonometry

A. Absolute Value of a Complex NumberPress the MODE key and selectRECTANGULAR for the ComplexFormat. In Example 1 of Section 9.1 ofthe textbook, it was shown that

1323 =+ i . Use the absolute value

function to check this. This and othercomplex number commands are underthe MATH\Complex menu.

E. Converting Rectangular to Polar CoordinatesRefer to Example 2b in Section 6 of the textbook where the point

)5,3(),( =yx in rectangular coordinates was converted to polar coordinates

)rad03.1,34(),( =θr . Put your calculator in radian mode, and executethe conversion commands as shown on the home screen to obtain the r and_ coordinates separately. These commands are under the MATH\Anglemenu. Note that a decimal point was inserted after the 5 in the second lineso the calculator would give an approximate angle.

F. Polar GraphingRefer to Example 6 in Section 6 where the polar curve θ2sin=r was graphed. Put your calculator in radianmode and in polar graphing mode by selecting MODE\Graph\POLAR. Press Y= to see the polar functionscreen. The calculator expects each r to be a function of _. The _ symbol can be found under the CHAR\Greekmenu. The WINDOW screen is similar to the one for parametric graphing: In addition to an x and y range, youmust specify a _ range that the calculator will use to plot polar points. For this example of graphing θ2sin=r ,use the range πθ 20 ≤≤ with a _ step of 20/π . Do zoom fit followed by zoom square.

Exercise: Trace the graph in the above example. In what order are the leaves of the rose traced?Exercise: Try a _ step of .001, and then try a _ step of .5. Explain what happened.


Chapter 11 – Systems of Equations

A. Linear System with a Unique SolutionInstead of using the reduced-row-echelon form command rref( to solve a linear system, you can use theSimultaneous Equation Solver flash application. This application can be downloaded from

http://education.ti.com/us/product/prselect.html. Suppose you want to solve the system

−=++

=−

−=++

21033

652

232

zyx

zy

zyx

given in Example 3 of Section 11.2 of the textbook. After you have installed this application, run it by selectinggreen diamond APPS\Simultaneous Eqn Solver\ENTER\New\ENTER. Enter the number of equations andunknowns as shown in the first screen. Enter the coefficients for the augmented matrix as shown in the nextscreen; maneuver using the ENTER key and the arrow keys. Press F5 (Solve) to solve this system.

Although the calculator uses the symbols K,,, 321 xxx for the unknowns, you should display your answer as

2,2,8 −=−== zyx .

Exercise: Solve the above system using the rref( command found under the MATH\Matrix menu.

B. Linear System with No Solution

In Example 4, Section 11.3 of the textbook, the system

=++

=++

=++

4753

2542

12

zyx

zyx

zyx

was shown to be inconsistent. We

will verify this using the simultaneous equation solver flash application described in the previous section. Asbefore, and the number of equations and unknowns are 3, the coefficients are entered as an augmented matrix.Press F5 (Solve) to verify that there are no solutions to this system.


C. Linear System with an Infinite Number of Solutions

In Example 6, Section 11.3 of the textbook, the system

=+−+

=+−

=+++

04023172

0642

0352

wzyx

wzy

wzyx

was found to be dependent

and the infinite solution was described parametrically. The simultaneous equation solver flash application willalso handle this case. Note that the number or equations and the number of unknowns do not have to be thesame. Although not visible on the middle screen, the last column of the augmented matrix consists of zeros forthis example.

To display the answer using the same style as your textbook, replace 1@ on the right side of the solution withthe parameter t. Thus, the solution is 0,,2,)2/11( ===−= wtztytx where t is understood to be any realnumber.


D. Solving a Nonlinear System

The nonlinear system

=

−=+−

3

12

xy

yx was solved in Example 1, Section

11.1.A of the textbook The solve command under the F2 (Algebra) menucan also be used to solve systems of equations. Enter

}),{,3and12(solve yxxyyx =−=+− on the command line. “and” can betyped using the alpha key or can be found under the MATH\Base menu; thebraces are above the parentheses keys. The complete solution

2,2/3 == yx and 3,1 −=−= yx can be seen by scrolling up to the outputand pressing the right arrow key.

Exercise: Check your answer by graphing.

E. Partial Fraction Decomposition

Refer to Example 7 in Section 11.2 of the textbook. Your calculator has thecapability of directly finding the coefficients A, B, and C so that

22

2

)2(21)2)(1(

10152

++

++

−=

+−

++

x

C

x

B

x

A

xx

xx. Execute the expand( command

which is found under the F2 (Algebra) menu to see that 3=A , 1−=B , and4=C .


Chapter 12 – Discrete Algebra

A. Table and Graph of Sequence ValuesRefer to Example 4 in Section 12.1 in the textbook. Instead of using thecalculator function y1 to create a table of values for the sequence

32 −−= nnan , we will use the sequence feature of the calculator. Select

MODE\Graph\SEQUENCE to put your calculator in sequence mode. PressY= to see the sequence functions u1, u2, u3… Type in the formula for thesequence as shown after u1; press alpha\N to get the n symbol. (The ui1symbol will be explained in Section B. below.)

Press WINDOW and set nmin=1 because the sequence na starts with the index 1=n . The values of the other

window variables will not matter when setting up a table. Set up the table to start with 1 and step by 1, and thenpress TABLE to see the sequence values.

Part 3 - TI-89/Voyage 200/TI-92+facstaff.cbu.edu/wschrein/media/M231 Notes/TI89manual.pdf · Part 3 - TI-89/Voyage 200/TI-92+ Chapter 1 – Basics A. About the TI-89, Voyage 200,

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