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

Transcription

1 Part 3 - TI-89/Voyage 00/TI-9+ Chapter 1 Basics A. About the TI-89, Voyage 00, TI-9+, TI-9 All of these calculators have symbolic algebraic capability. This means that they can solve equations and do calculus using letters. Therefore, they may not be allowed in certain classes or on certain exams. The TI-89 is about the same size as the TI-83+, and may be more acceptable because it does not have a QWERTY keyboard like the TI-9+ and Voyage 00. The TI-89, 9+ and Voyage 00 can run flash applications and have their operating systems upgraded. These can be downloaded from The TI-89 and Voyage 00 are powerful, but it takes about twice as many button presses to do the same task as it does on a TI-83+. Hence, some students with TI-89s have trouble following an instructor who is using a TI The TI-89 and above also have unit conversions and physical constants built in. The TI-9 cannot run flash applications nor have its operating system upgraded like the TI-89 and Voyage 00. However, most of the instructions given in this section of the manual will also work for the TI-9. B. General Commands In addition to the obvious purpose of the ON key for turning on the calculator, it will also halt the execution of a 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 end of the manual that came with your graphing calculator. The screen that you see when first turn on the calculator is the home screen; this is the screen where most calculations are done. All functions in yellow on the face of the calculator are obtained by pressing the nd key and the desired key. The characters in purple are obtained by pressing the alpha key and the desired key. All functions in green on the face of the calculator are obtained by pressing 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 the catalog by pressing CATALOG. For example, if you are looking for the rref( command, you can do R to quickly move to the commands starting with the letter R. Then use down arrow key to select rref( and then press ENTER. C. The MODE key Press the MODE key and use the arrow and ENTER keys so that the settings are as shown in the picture. Press F or F3 to see the other two pages of modes. These are the usual settings, but there will be a few that you may want to change later. Press the ENTER key to save the changes you made and to return to the home screen. Page 1 Page Page 3 D. Entering Expressions Algebraic expression evaluation obeys the Please Excuse My Dear Aunt Sally rule. In decreasing order of priority, the operations are 1) Parentheses

2 Sally rule. In decreasing order of priority, the operations are ) Exponentiation 3) Multiplication and Division 4) Addition and Subtraction See your calculator manual for a more specific description of the order of evaluation. Note that there are separate buttons for subtraction and negation (-). If you are ever in doubt as to the how your calculator will evaluate an expression, then use parentheses. The ENTRY and ANS keys are useful for doing complicated calculations. The ENTRY key retrieves a previous expression, while the ANS key retrieves a previous numerical result. Alternatively, you can up arrow into the history screen and press ENTER to bring a desired expression or value to the command line. Note that an expression can be edited using the delete DEL, insert INS, and arrow keys. Pressing ENTER will cause an expression to be evaluated or command to be executed. Scientific notation is expressed using the EE key. Note that although this key shows two Es, only one is displayed on the screen. When you write results on paper, you should use standard notation, not calculator notation. For example, the number on the command line shown here should be 1 written as 3 10 on paper. Back to the Table of Contents Chapter Graphs and Technology A. Basic Graphing Press the Y= key and enter the right side of the equation y = x 3 x after y1. Select F1 (Tools)\9:Format and make sure the settings are as shown here. These are the settings that will be appropriate for most graphing. Press ENTER to save your settings. Select F (Zoom)\ZoomDecimal and a graph should appear. Press WINDOW to see the x and y ranges corresponding to zoom decimal and change xres to 1 or. You can press GRAPH to see the graph again. Your calculator graphs by plotting points and connecting the points with line segments. If you look carefully at the calculator screen when your calculator is turned on, you might see a grid of tiny rectangles. Each rectangle is either turned on (made dark) or turned off (made light) to make a graph. xres is the number of pixels that your calculator skips in the horizontal direction when plotting the endpoints of the line segments. xres can range from 1 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.

3 B. A Good Graph A good graph should definitely clearly show the structure that you are interested in. For example, if you are solving for where an expression is zero, the corresponding x intercept should be clearly visible. It is usually possible 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 zoom decimal window chosen in Example A was a good choice for graphing the equation y = x 3 x. Exercise: Try to find a good window that will show all the important features of y = x 3 1x + 0x. Then, find all the x-intercepts by factoring. Did your good graph clearly show all the x-intercepts? C. Missing Parts of Graphs Sometimes your calculator leaves gaps in a graph. This can happen when an equation is defined on one end and undefined at the other of a line segment that your calculator is trying to plot. For example, graph y = 6 x in the window shown here. The graph should be a complete semicircle, but your calculator leaves off the ends. When you copy a graph to paper, you should fill in details that your calculator omits. Exercise: Do zoom square so that the semicircle is not stretched out. Graph given by your calculator Graph as it should appear D. Graphing Asymptotes Sometimes your calculator will graph vertical lines that are not part of the graph. This can happen when a graph blows up to positive infinity on one side of the line and negative infinity on the other side. Such lines are called vertical asymptotes, and since they are not part of the graph, they should be graphed as dashed lines. For example, graph the equation y = in the window x 3 shown here. See that your calculator graphs the 3 vertical line x =. This line should be drawn as dashed when transferring the graph to paper. Graph as drawn by calculator Graph as it should appear

4 Exercise: Redo the example in Example D using a zoom decimal window. (Even though your calculator does not graph the vertical line, you should still graph it as a dashed line on paper.) E. Showing an Equation Is an Identity Exercise 41 in Section.1 of the textbook explains how to verify if an equation is an identity or not by graphing both sides of the equation. Instead of graphing the sides separately and tracing, you can graph only y 1 y + 1. The equation is an identity if and only if the graph of y 1 y + 1 is the horizontal line y = 1. (Can you explain why?) Before you graph, deselect y1and y. The y1 and y symbols used for y3 are simply typed using the y key and the numbers 1 and. Here is the technique applied to show that ( x + 5) = x + 10x + 5 is an identity. A zoom decimal window was used. Since the graph is the horizontal line y = 1, we are fairly confident that this equation is an identity. F. Showing an Equation Is Not an Identity Here is an example using the technique given in Example E showing that ( x + 5) = x + 5 is not an identity. A zoom decimal window was used. Since the graph is a not horizontal line, we are certain that this equation is not an identity. G. Linear Regression Refer to Example in Section.5 of the textbook. Although there is a data editor built into the TI-89 and Voyage00, the 83+ style statistics/list editor application is easier to use and more powerful. Also, it is easier to follow your teacher if he or she is using a TI-83+. This application can be downloaded from TI s website After you have installed this application, run it by selecting green diamond APP\Stats/List Editor\ENTER. We will use L1 and L for the names for the x and y lists, respectively. Note that the TI-89 and Voyage 00 do not distinguish upper and lowercase letters. Clear out all functions under the Y= key. To set up a scatter diagram, you can press ENTER over the Plot 1 symbol in the Y= menu. Then select the scatter diagram icon, and select lists L1 and L as shown in the picture; press ENTER 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.

5 To find the least-squares regression line, go back to the statistics/list editor application. Select F4 (Calc)\Regressions\LinReg(ax+b). Then fill in the dialog box so the calculator knows you are using lists L1 and L, and you want the regression equation stored in y1(x). You should write the regression equation on paper as y =. 43x ; do not just write the values of the coefficients a and b. Press Y= to verify that the regression equation is in y1. Then press GRAPH to see the regression equation graphed with the scatter diagram. Suppose you want to predict the poverty level in 005. Since the regression equation is already in y1, go to the home screen and evaluate this function at 15. (The TI-89 and Voyage use function notation that will be explained in Chapter 3, Section A.) Thus, the predicted poverty level is $19,753. Prediction using the home screen Exercise: Predict the poverty level in 005 graphically by doing GRAPH]ZoomOut\ENTER and then F5 (Math)\Value\15\ENTER. Back to the Table of Contents Prediction using the graph screen Chapter 3 Functions and Graphs A. Evaluating a Function We will evaluate f ( x) = x + 1 using function notation for the x value of 3 and the x value of - 5. First, put this function in y1, and then go to the home screen. Note that including a decimal point in the expression caused the calculator to give approximate values for f ( 3) = 10 and f ( 5) = 6. This could also have been accomplished by pressing green diamond ENTER.

6 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, press TblSet and set the parameters as shown in the screen below. Press TABLE and try scrolling through the table using the up and down arrow keys. Graph the greatest integer function in a zoom decimal window. If your calculator is in connected mode, then it will graph vertical parts of the steps that are really not part of the graph. Go to the Y= screen, press F6(Style)\Dot, and then press GRAPH again to obtain a more correct looking graph. Try tracing your graph to see 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 x if x 1 The piecewise-defined function f ( x) = was graphed in Example 6 of Section 3.3 of the x + if 1 < x 4 textbook. To graph it on your calculator, type the function in y1 as when(x 1, x, when(x 4, x +, undef)). The symbol can be obtained by typing green diamond <, when can be found in the CATALOG or typed using the alpha key, and undef must be typed using the alpha key. Use the window values shown below and graph in dot mode. Note that f (x) is not defined if x > 4.

7 D. Minimum of a Function 3 This example refers to Example 8 in Section 3.3. Enter the function f ( x) = x 1.8x + x + 1 in y1 and try to find a window that clearly shows the extrema without looking at the graph range shown here. Tracing will only give points that are within a pixel of the true minimum. To find and accurate value for the minimum, press GRAPH\F5(MATH)\Minimum and answer the prompts using the left and right arrow keys and pressing ENTER. The minimum value for the function f is y = and it occurs at x = E. Parametric Graphing This example refers to Section 3.3 Example 13. Press MODE\Graph\PARAMETRIC\ENTER to put your 3 calculator in parametric mode and then enter the functions x = t t 1 and y = t 4t 6 ( t 3) under the Y= menu. Press WINDOW and enter the values shown below. You can easily find a good window by doing F(Zoom)\ZoomFit. F3(Trace) the graph to see how the ( x, y) points correspond to the t parameter values. Exercise: Try different tstep values in the previous example. What happens if tstep is too small? What happens if tstep is too large? You can also make a table showing the points that your calculator is plotting to create the parametric graph. Adjust the values under TblSet as shown in the first screen, and press TABLE to see the second screen. F. Power Functions x n This example explores the shape of the power functions by putting a list in a function. Enter the functions as shown in the first screen, and deselect y. Then graph in a zoom decimal window. This will graph 3 x, 5 x, and 7 x simultaneously. Notice that all these graphs are symmetric with respect to the origin. In fact, all power functions with odd exponents are odd functions.

8 Now select y1 and deselect y. Then graph in a zoom decimal window. This 4 6 will graph x, x, and x simultaneously. Notice that all these graphs are symmetric with respect to the y-axis. In fact, all power functions with even exponents are even functions. G. Vertical Shift Put the square root function in y1 and select a thick line plotting style. Add 4 to the outside of the function y1 to shift it up 4 units. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range. A table can also be constructed to demonstrate a vertical shift. Press MODE\Complex Format\REAL\ENTER so that x will be undefined for negative values instead of returning nonreal values. Note that all the y values of y in the table are 4 greater than the y values of y1 H. Horizontal Shift Put the square root function in y1 and select a thick line plotting style. Add to the inside of y1 to shift it to the left units. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range. A table can also be constructed to demonstrate a horizontal shift. Note that all the y values in the y1 column are shifted units up to get the y values of y column.

9 I. Vertical Expansion Put the square root function in y1 and select a thick line plotting style. Multiply the outside of y1 by to expand it vertically by. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range. A table can also be constructed to demonstrate a vertical expansion. Note that all the y values in the y column are twice as large as the y values in the y1 column. J. Vertical Contraction Put the square root function in y1 and select a thick line plotting style. Multiply the outside of y1 by 0.5 to contract it vertically by 0.5. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range. A table can also be constructed to demonstrate a vertical contraction. Note that all the y values in the y column are half as large as the y values in the y1 column. K. x-axis Reflection Put the square root function in y1 and select a thick line plotting style. Negate the outside of y1 to reflect it about the x-axis. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range.

10 A table can also be constructed to demonstrate an x-axis reflection. Note that all the y values in the y column negatives of the y values in the y1 column. L. y-axis Reflection Put the square root function in y1 and select a thick line plotting style. Negate the inside of y1 to reflect it about the x- axis. Use a zoom standard window to obtain the graph shown here. Exercise: Trace y in order to estimate its domain and range. A table can also be constructed to demonstrate a y-axis reflection. Note how the y values in the y column are related to the y values in the y1 column. M. Composition This example refers to Example 5 in Section 3.5. Put f ( x) = x in y1 and g ( t) = t 5 in y and deselect both of these functions. The composition g o f is shown in y3. Graph y3 in the window shown below. Note that the graph agrees with the rule for the composition ( g o f )( x) = x + 5, x 0 that was derived in the textbook. Exercise: Find the domain and range of the function N. Instantaneous Rate of Change g o f. 3 x Refer to Example 9 of Section 3.6 of the textbook; the volume of a balloon is given by V ( x) = in gallons 55 where x is the radius of the balloon in inches. A table can be used to find the instantaneous rate of change of V when x = 7 inches. Put the formula for the volume of the balloon in y1 and the difference quotient V ( 7 + h) V (7) in y. Set the independent variable to Ask under TblSet and enter decreasing values of h for x h

11 under TABLE as shown below. As x converges to zero, the difference quotient converges to.673. Thus, the rate of change of the volume at the instant x = 7 inches is.673 gallons/inch. O. Inverse Function Put the function ( ) = x 3 3 f x + 5 in y1 using a thick graphing style, the function g ( x) = x 5 in y, and the identity function in y3. The first graph was done using a zoom standard window. Zoom square was applied again to obtain the second graph so that the line y = x would appear diagonal. Note that the functions f and g are reflections of each other about this line because they are inverse functions. If you are starting with only f ( x) = x 3 + 5, then you can easily graph its inverse by executing the GRAPH\F6(Draw)\DrawInv y1(x) command on the home screen. An inverse function cannot be traced when graphed using the draw inverse command. Another way to graph both a function and its inverse is to put your calculator in parametric mode. The first parametric function contains f, the second contains g, and the third contains the identity function. First do zoom standard in order to adjust the x and y window ranges, and then use the t range shown here. Lastly, do zoom square. Back to the Table of Contents

12 Chapter 4 Polynomial and Rational Functions A. Evaluating Polynomials The usual way of writing a polynomial is to write it as a sum of monomials with decreasing powers. For 4 3 example, the polynomial f ( x) = 5x + x + 7x + 7x + 8 is written in the standard form. Your calculator will evaluate a polynomial faster if it is written in nested form. For example, f ( x) = ( ( 5x + ) x + 7) x + 7) x + 8 is the same polynomial written in nested form. After some practice, you can quickly write down the nested form of a polynomial. The derivation of the nested form involves successively factoring out powers of x. Here is a derivation for this polynomial: f ( x) = = = = 3 ( 5x + ) x + 7x + 7x ( 5x + ) x + 7x + 7x + 8 ( 5x + ) x + 7) x + 7x + 8 ( ( 5x + ) x + 7) x + 7) x + 8 The reason your calculator will evaluate a polynomial more quickly in nested form is multiplication is performed faster than exponentiation. For the polynomial given here, the standard form has 3 exponentiations and 4 multiplications; the nested form has only 4 multiplications. Exercise: Test that the nested form of a polynomial is evaluated quicker on your calculator than the standard form. Put the standard form of the polynomial f (x) given in the above example in y1 and the nested form in y, and deselect y. Set xres=1, find a good window, and then time how long it takes your calculator to graph the standard form. Next, deselect y1, select the nested form in y, and time how long it takes the nested form to graph 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 00 for evaluating polynomials. The polynomial f (x) would look like polyeval({5,,7,7,8,}, x ) on your calculator. The polynomial evaluator command is under the CATALOG, and the braces are above the parentheses keys. The coefficients for the polynomial appear in the braces separated by commas, so do not forget to put zeros in for any missing coefficients. The last argument of the polyeval is where you want to evaluate the polynomial. Exercise: Test if the polyeval Command graphs faster than the nested form. B. Polynomial Division The TI-89 and Voyage 00 can symbolically divide polynomials. For 4 3 example, suppose you want to divide f ( x) = 5x + x + 7x + 7x + 8 by g ( x) = x + x + 5. First do F6\(Clean Up)\Clear A-Z in order to delete the calculator variable x. Then perform the following operations on the home 5x^4 + x^3 + 7x^ + 7x + 8 x^ + x + 5 g f

13 calculator variable x. Then perform the following operations on the home screen: 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 as shown on the screen. This command is under the F (Algebra) menu. Thus, the quotient is 5x 3x 15 and the remainder is 37 x Note that you could also write the relationship between these polynomials as f ( x) = 5x 3x 15 g( x) + 37x ( ) ( ) C. Polynomial Solver Although the TI-89 and Voyage00 do not have a polynomial solver built in, but a free flash application that finds the approximate zeros of polynomials can be downloaded from TI s website Suppose you want to solve the equation 4x 1x + 8x 1 = 0 given in Example 5 of Section. in the textbook. After you have installed this application, run it by selecting green diamond APP\Polynomial Root Finder\ENTER\New The application first prompts for the degree of the polynomial. After typing in 5, press ENTER to go to the next screen where the coefficients of the polynomial are entered. Note zeros are entered for the missing coefficients. Press F5 (Solve) to find the approximate roots. Note that this polynomial equation has 5 distinct real roots. D. Quick Bounds for Zeros and Graphing Polynomials There is a faster, although less accurate test for bounding the zeros of a polynomial than the Bounds Test introduced in Section 4.3 of the textbook. In general, if z is a zero of the polynomial n n 1 max( an 1, an, L a1, a0 ) f ( x) = an x + an 1x + L + a1x + a0 where a n 0, then z + 1. For a example, consider the polynomial f ( x) = x 6x + 9x + 7x 8x + 33x 36x + 0. The largest the max( 6, 9, 7, 8, 33, 36, 0) absolute value of a zero for f can be is + 1 = Thus, every real zero of f is guaranteed to be in the interval 37 x 37. This can be used to start looking for a good graph that will not omit any x intercepts. Enter this polynomial in y1. (You do not have to use the polyeval command.) Use the graph window shown here. n

14 This first graph shows that all the real zeros are in the interval 5 x 5. Use this for the x range and do zoom fit. Adjust the window again to get a good graph. Exercise: Do zoom box in order to see the positive x-intercepts more clearly. E. Horizontal Asymptotes x Define the rational function f ( x) =. In Example 5 in Section 4.3 of the textbook, it was shown that x + x y = is the horizontal asymptote for f. See the screens below to see how to graph this rational function with its horizontal asymptote. (The function y1 has a thick-line graphing style.) You can also see that y = is a horizontal asymptote by making a table. Set up the table so that it asks for the independent variable. Then you can see that as x approaches either infinity or negative infinity that f (x) approaches. Exercise: Why is y1 undefined when when x = 1 in the above table?

15 F. Oblique Asymptotes In Example of Section 4.5.A, it was shown that y = x + 3x 4 is the oblique asymptote for the function 3 x + x 7x + 5 g ( x) =. The screens below show how to graph g with its oblique asymptote. (Give y1 a thickline plotting x 1 style.) This can also be demonstrated with a table. Deselect y1 and y and put the difference in y3. Set up the table so that it asks for the independent variable. Then you can see that as x approaches either infinity or negative infinity, that the difference between f (x) and its oblique asymptote approaches zero. Exercise: Why is there an error when x = 1 in the above table? G. Complex Arithmetic Press the MODE key and select Complex Format\RECTANGULAR so that your calculator will do complex numbers and do them in standard form. Your calculator will do addition, subtraction, multiplication, division, and exponentiation of complex numbers. They are typed on the home screen just like they appear in the textbook. The complex constant i is above the CATALOG key on your calculator. This example shows how to multiply two complex numbers, raise i to a power, and divide two complex numbers. Back to the Table of Contents

16 Chapter 5 Exponential and Logarithmic Functions A. Root Command n 1/ n The identity x = x must be used to compute a root. The screen shown to the right demonstrates taking the 4 th root of 81. Exercise: Graph the functions y = x, y = 3 x, y = 4 x, and y = 5 x in a zoom decimal window. How does the shape of the graph, the domain, and the range of y = n x depend on n? B. Exponential and Logarithmic Functions x In this example, we will graph the exponential function y = 10, and its inverse function, the common logarithm y = log10 x 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 are x reflections of each other about the diagonal line y = x. Trace y1 to see that y = 10 does not touch the x-axis. Also, y = log10 x does not touch the y-axis. Exercise: What is the domain and range x of y = 10? What is the domain and range of y = log10 x? Do your answers agree with the graph? x Exercise: Graph y = e and y = ln x with y = x. Then write down the domains and ranges for y = ln x and relate them to your graph. x y = e and C. Nonlinear Regression Regression curves can also be found for the models studied in Chapter 5. The first column in the table shown at the right gives the names of the models, the second column gives the formulas for the curves, and the third column gives the calculator commands for finding those regression curves. These commands are under the STAT\CALC menu and the syntax is the same as for linear regression. Refer to Example 1 in Section 5.6 of the textbook. Enter the data in the lists L1 and L using the statistics editor flash application. Then set up a scatter diagram and do zoom data. Notice that the shape of the scatter diagram suggests that an exponential model is appropriate. model logarithmic exponential power logistic like on the TI-83 logistic like on the TI-86 formula calculator command y = a + bln x LnReg x y = ab ExpReg b y = ax PowerReg c y bx + ae Logistic83 a y = d cx + be 1 Logistic

17 Select F4(Calc)\Regressions\ExpReg on the home screen to find that the best-fitting exponential regression y = x. curve is ( ) Back to the Table of Contents Chapter 6 Trigonometric Functions A. Degrees and Radians A common mistake when evaluating trigonometric functions is to be in the wrong angle mode. An angle mode can be set by pressing the MODE key and then selecting DEGREE or RADIAN after Angle. Press ENTER to save your choice and return to the home screen. Note that your calculator displays the angle mode at the bottom of the home screen. Setting the mode to degrees Setting the mode to radians The angle mode is indicated by DEG or RAD at the bottom of the screen. B. Cosecant, Secant, Cotangent These functions come with operating system version.08 for the TI-89 and Voyage 00, but there is a bug that TI has not fixed yet involving them. Therefore, it is recommend that you use the reciprocal identities csc t =, sec t = and cot t = with the SIN, COS, and TAN sin t cost tan t keys, respectively, instead. For example, if you wanted to approximate cot 10, then put your calculator in degree mode, and execute the statement as shown on the home screen as shown here.

18 C. Evaluating Trigonometric Functions Use parentheses when in doubt as to how your calculator will evaluate an expression. For example, the next screen shows how to evaluate sin x at x = 1.9 radians. Do not forget to put your calculator in radian mode first! The first screen shows two correct ways to evaluate sin x at x = 1.9 radians. (Your calculator does not understand the power notation sin x, so you will have to use the definition sin x = (sin x).) The second screen shows a wrong way to compute sin x. Correct value of x sin This is ( ) sin x, not sin x. D. Zoom Trig Window for the TI-89 This is a convenient window to use when graphing trigonometric functions. Although graphing is usually done in radian mode, keep in mind that the x range depends on the angle mode. After selecting radian or degree 1 mode, press ZOOM\ZoomTrig. If your calculator is in radian mode, then the x range is approximately 3 π 4 1 to 3 π with an x scale of π /. If your calculator is in degree mode, then the x range is approximately to 590 with an x scale of 90. Radian zoom trig window Degree zoom trig window y = sin t graphed using zoom trig Exercise: Approximately how may periods of y = sin t appear in the zoom trig window shown above? D. Zoom Trig Window for the Voyage 00 This is a convenient window to use when graphing trigonometric functions. Although graphing is usually done in 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 5π to 5 π with an x scale of π /. If your calculator is in degree mode, then the x range is approximately 900 to 900 with an x scale of 90.

19 Radian zoom trig window Degree zoom trig window y = sin t graphed using zoom trig Exercise: Approximately how may periods of y = sin t appear in the zoom trig window shown above? E. Amplitude, Period and Phase Shift We will redo Example 1 in Section 6.5.A of the textbook in more detail. In that example, the constants A, b, c were estimated where Asin( bt + c). = 4sin(3t + ) + cos(3t 4) Graph the right side of this equation using the window shown here. Use your calculator s maximum command to find that the amplitude is Press HOME and store the x coordinate of this maximum in the calculator variable A. Find the next maximum and return to the home screen. Then subtract the two x values to find that the period is π π.09. Solve period = for b to get b =. Ignoring round-off error, the calculator agrees the correct b period value of b = 3. 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 y = 3.94sin(3t) in y and press GRAPH. y can be shifted either left or right to obtain y1, but we will assume the phase shift is negative because this magnitude of shift is less. Find the maximum of y as shown and subtract its location from the location of the original maximum found for y1 to find that the phase shift is Note that the phase shift is negative because we decided that y1 should be shifted left to obtain y.

20 c Solve phase shift = for c to get c = b( phase shift). Use this last b equation to find that c =. 51. Thus, 4 sin(3t + ) + cos(3t 4) = 3.94sin(3t +.51). F. Sine Regression Refer to Example 11 in Section 6.5 of the textbook. Enter the month data in list L1 and the temperature data in list L in the statistics editor flash application. Do not forget to duplicate the data so that the months run from 1 to 4. Then create a scatter diagram by doing zoom data. Sine regression is appropriate because the shape is a sine wave. In the flash statistics editor, sine regression command is under the F4\CALC\Regressions\SinReg menu. Fill in the dialog box as shown below and press ENTER. Press the right arrow key to see the rest of the regression coefficients. You should write the regression equation as y =.7 sin(0.5t.18) on paper. Press the GRAPH key to see how the regression equation fits the scatter diagram. Exercise: Why is the period in the above example equal to 1? Back to the Table of Contents Chapter 7 Trigonometric Identities and Equations

21 A. Showing an Equation is Not an Identity Refer to Example 1(a) of Section 7.1 in the textbook. One way to check if an equation is an identity is to graph both sides of the equation. For example, to determine if the equation sin x cos x = cos x + sin x is an identity, put the left side in y1 and the right side in y. 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. Since the graphs of both sides of the equation are not the same, the equation is not an identity. To show an equation is not an identity, it is also sufficient to find a single value for which both sides are different. The table feature can be used to also see that both sides are not equal. B. Showing an Equation is an Identity Refer to Example 1(a) of Section 7.1 in the textbook. We will use the technique demonstrated in Section A to 1+ sin x sin x show that = cos x + tan x is an identity. Note that both graphs are exactly on top of each other cos x 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 the functions are the same. Note that we have not proven that this equation is true for all x. We only have strong statistical 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, you need only graph y 1 y + 1. The equation is an identity if and only if the graph of y 1 y + 1 is the horizontal line y = 1. (Can you explain why?) Before you graph, deselect y1 and y.

22 C. Checking an Exact Answer In Example 5(a) of the textbook, it was shown that the exact value of 5π cos is. To check this, ensure that your calculator is in 8 AUTO mode. Note that a large number of decimal places does not mean the number is exact. D. Graphing an Inverse Trigonometric Function A graph of sin x in a ZoomTrig window shows that the sine function is not one-to-one. π π You can graph the sine function restricted to x by putting when( x π / and x π /,sin( x),undef ) in y1. The when command is under the CATALOG; the inequalities can be obtained by doing green diamond > and green diamond >; the and symbol is under the MATH\BASE menu; just type the undef symbol using the alpha key. Use ZoomBox to get a larger graph of the restricted sine. Note that this function is one-to-one. Select the thick line style for y1 and put the identity function in y. To graph 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 inverse sine reflected across the line y = x. Note that you cannot trace an inverse function when it is drawn in this manner. Exercise: Redraw the inverse sine function using ZSQR. Exercise: Draw the inverse sine function using the SIN -1 button on your calculator.

23 E. Composing a Trigonometric and Inverse Trigonometric Function In section 7.4 of the textbook, it is shown that sin 1 sin u = u if π π u. This is demonstrated using the technique shown in Section A. Note that sin 1 sin u = u by itself is not an identity. In section 7.4 of the textbook, it is shown that sin sin 1 v = v if 1 v 1. This is demonstrated using the technique shown in Section A. Note that sin sin 1 v = v is an identity. F. All the Solutions of a Trigonometric Equation Refer to Example 1 in Section 7.5 of the textbook where all solutions to tan x = were found. Both sides of the equation were graphed using an x range of π x 4π and a y range of y 4. Note that the intersection of the vertical asymptotes and the line y = do not give solutions to the equation tan x =. Use the Intersection command under the GRAPH\Math menu to find the smallest positive intersection of y = tan x and y =. Since the tangent function has period!, all the relevant roots are separated by!. Thus, the solution set is given by x = kπ where k is an arbitrary integer. You can verify that this formula gives solutions to the equation tan x = by using the table feature of your calculator:

24 Use the solve( command that is found under the F (Algebra) menu to directly find all the roots of the equation tan x =. First, ENTER was pressed giving the exact solution in terms of the inverse tangent function. Then, green diamond was pressed to give the approximate solution that we had found graphically. symbol represents an arbitrary integer. G. Checking an Identity in Two Variables We will graphically show that cos( x y) = cos( x)cos( y) + sin( x)sin( y) is an identity using a technique similar to the one shown in Section B. Since this equation involves two variables (x and y), our graphs will involve three variables (x, y, and z) and will be surfaces in space. Put your TI-89 in 3-dimensional graphing mode by doing MODE\Graph\3D. Press Y= to see the 3-D function editor. Put the left side in z1 and the right side in z, and the difference in z3. Deselect z1 and z. Then press F1\Format and ensure that the settings are the same as shown here. Press WINDOW and set the x and y ranges from π to π ; set the z range from - to. Press GRAPH to see the graph of z1-z as a function of ( x, y). Since the graph is the plane z = 0, this confirms that z 1 = z. Try tracing and pressing the arrow keys to move around the grid and see that all the z s are approximately zero. Back to the Table of Contents 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 of a = 16 A = 44. an oblique triangle correctly. Refer to Example 1 in Section 8.3 of the textbook. After b = 10 B = 5.8 solving the triangle, all the parts were found as shown here. c = 1.6 C = 110 Every triangle satisfies Mollweide s Equation: C ( a b)cos A B = csin

25 If your values for the sides and angles satisfy this equation, then it is likely that you have not made an error. Likewise, if your values for the sides and angles do not satisfy this equation, then it is likely that you have made an error. First, put your calculator in degree mode. Then evaluate the both sides of Mollweide s Equation as shown in the calculator screen. Both sides agree to roughly three significant digits, confirming that the sides and angles are correct. Exercise: Why do both sides of Mollweide s Equation only agree to three significant digits in the above example? B. Solving an Oblique Triangle The triangle having the three parts a = 7. 5, b = 1 and A = 35 was solved in Section 8.3, Example 4 of the textbook. In this example, it will be shown how to solve this triangle by doing a chain calculation on the home screen. This is an example of the SSA case where there are two possible triangles that satisfy the given sin B sin 35 information. Solve the Law of Sines formula = to find sin B = (Although only 3 significant digits are being written down in this paragraph, 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 be subscripted with a 1, and the parts of the second triangle will be subscripted with a. Apply the inverse sine to find the angle B 1 = This is stored in the calculator variable B because it will be needed for the second triangle. Since the sum of the angles in a triangle is 180, find sin 78.4 sin 35 C 1 = 180 A B1 = Solve = to find c 7.5 a sin C c 1 1 = = 1.8. It remains to solve the second triangle. Take the sin A supplement to find B = 180 B1 = 113. Since the sum of the angles in a triangle is 180, find C = 180 A B = Redo the previous Law of Sines formula to get c = Back to the Table of Contents Chapter 9 Applications of Trigonometry A. Absolute Value of a Complex Number Press the MODE key and select RECTANGULAR for the Complex Format. In Example 1 of Section 9.1 of the textbook, it was shown that 3 + i = 13. Use the absolute value function to check this. This and other complex number commands are under the MATH\Complex menu. 1

26 E. Converting Rectangular to Polar Coordinates Refer to Example b in Section 6 of the textbook where the point ( x, y) = (3,5) in rectangular coordinates was converted to polar coordinates ( r, θ ) = ( 34, 1.03 rad). Put your calculator in radian mode, and execute the conversion commands as shown on the home screen to obtain the r and _ coordinates separately. These commands are under the MATH\Angle menu. Note that a decimal point was inserted after the 5 in the second line so the calculator would give an approximate angle. F. Polar Graphing Refer to Example 6 in Section 6 where the polar curve r = sin θ was graphed. Put your calculator in radian mode and in polar graphing mode by selecting MODE\Graph\POLAR. Press Y= to see the polar function screen. The calculator expects each r to be a function of _. The _ symbol can be found under the CHAR\Greek menu. The WINDOW screen is similar to the one for parametric graphing: In addition to an x and y range, you must specify a _ range that the calculator will use to plot polar points. For this example of graphing r = sin θ, use the range 0 θ π with a _ step of π / 0. 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. Back to the Table of Contents Chapter 11 Systems of Equations A. Linear System with a Unique Solution Instead of using the reduced-row-echelon form command rref( to solve a linear system, you can use the Simultaneous Equation Solver flash application. This application can be downloaded from x + y + 3z = Suppose you want to solve the system y 5z = 6 3x + 3y + 10z = given in Example 3 of Section 11. of the textbook. After you have installed this application, run it by selecting green diamond APPS\Simultaneous Eqn Solver\ENTER\New\ENTER. Enter the number of equations and unknowns as shown in the first screen. Enter the coefficients for the augmented matrix as shown in the next screen; maneuver using the ENTER key and the arrow keys. Press F5 (Solve) to solve this system.

27 Although the calculator uses the symbols x 1, x, x3, K for the unknowns, you should display your answer as x = 8, y =, z =. Exercise: Solve the above system using the rref( command found under the MATH\Matrix menu. B. Linear System with No Solution x + y + z = 1 In Example 4, Section 11.3 of the textbook, the system x + 4y + 5z = was shown to be inconsistent. We 3x + 5y + 7z = 4 will verify this using the simultaneous equation solver flash application described in the previous section. As before, 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. Exercise: Solve the above system using the rref( command found under the MATH\Matrix menu. C. Linear System with an Infinite Number of Solutions x + 5y + z + 3w = 0 In Example 6, Section 11.3 of the textbook, the system y 4z + 6w = 0 was found to be dependent x + 17y 3z + 40w = 0 and the infinite solution was described parametrically. The simultaneous equation solver flash application will also handle this case. Note that the number or equations and the number of unknowns do not have to be the same. Although not visible on the middle screen, the last column of the augmented matrix consists of zeros for this example.

28 To display the answer using the same style as your textbook, 1 on the right side of the solution with the parameter t. Thus, the solution is x = ( 11/ ) t, y = t, z = t, w = 0 where t is understood to be any real number. Exercise: Solve the above system using the rref( command found under the MATH\Matrix menu. D. Solving a Nonlinear System x + y = 1 The nonlinear system was solved in Example 1, Section xy = A of the textbook The solve command under the F (Algebra) menu can also be used to solve systems of equations. Enter solve( x + y = 1 and xy = 3,{ x, y}) on the command line. and can be typed using the alpha key or can be found under the MATH\Base menu; the braces are above the parentheses keys. The complete solution x = 3 /, y = and x = 1, y = 3 can be seen by scrolling up to the output and pressing the right arrow key. Exercise: Check your answer by graphing. E. Partial Fraction Decomposition Refer to Example 7 in Section 11. of the textbook. Your calculator has the capability of directly finding the coefficients A, B, and C so that x + 15x + 10 A B C = + +. Execute the expand( command ( x 1)( x + ) x 1 x + ( x + ) which is found under the F (Algebra) menu to see that A = 3, B = 1, and C = 4. Back to the Table of Contents Chapter 1 Discrete Algebra A. Table and Graph of Sequence Values Refer to Example 4 in Section 1.1 in the textbook. Instead of using the calculator function y1 to create a table of values for the sequence = n n 3, we will use the sequence feature of the calculator. Select a n MODE\Graph\SEQUENCE to put your calculator in sequence mode. Press Y= to see the sequence functions u1, u, u3 Type in the formula for the sequence as shown after u1; press alpha\n to get the n symbol. (The ui1 symbol will be explained in Section B. below.) Press WINDOW and set nmin=1 because the sequence a n starts with the index n = 1. 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 then press TABLE to see the sequence values.