Contents 6. Systems of Linear Equations and Determinants 2 Example 6.9................................. 2 Example 6.10................................ 3 6.5 Determinants................................ 4 Evaluating Determinants on a Calculator.................. 4 Example 6.33................................ 4 1
Peterson, Technical Mathematics, 3rd edition 2 Example 6.9 A line passes through the point (2, 4) and makes an angle of 115 with the positive x-axis. What is the equation of the line? Solution The slope of the line is given tan 115. To approximate tan 115, make sure that your calculator is in degree mode and press 2nd TAN 115 ) ENTER. The result shown in Figure 6.6a is not very helpful. To get a numerical value, press ENTER [ ] and you get the result in Figure 6.6b. FIGURE 6.6a FIGURE 6.6b Notice in Figure 6.12c that the values of tan 65 and tan 115 differ only in their sign. You ll learn the reason for that when you study some more trigonometry in Chapter 10. Now that we have the slope, we can apply the point-slope form of the linear equation to obtain FIGURE 6.6c y = 4 + (x 2) tan 115 4 + (x 2)( 2.1445) This is y 2.1445x + 0.2890 when it is written in slope-intercept form. The graphs of y 2.1445x 8.289 and y 2.1445x + 0.2890 are in Figure 6.7a. Notice that they appear to be almost perpendicular. This is because we used the standard calculator s viewing window. If you select the ZoomSquare window by pressing F2 [ZOOM] 5 [ZoomSqr] you obtain the graphs in Figure 6.7b. You can also see them drawn on a coordinate system in Figure 6.7c in the textbook.
Peterson, Technical Mathematics, 3rd edition 3 FIGURE 6.7a FIGURE 6.7b
Peterson, Technical Mathematics, 3rd edition 4 Example 6.10 A line passes through the point (2, 4) and makes an angle of 65 with the positive x-axis. What is the equation of the line? Solution The slope of the line is tan( 65 ) and the approximate value of tan( 65 ) is shown in Figure 6.8. Notice that tan( 65 ) = tan 115. This is because the difference in the measures of their angles is 115 ( 65 ) = 180. Again, if we apply the angle-point form of the linear equation we see that the equation of this line is y 4+(x 2)( 2.1445) or, when written in slope-intercept form, is y 2.1445x + 0.2890. FIGURE 6.8
Peterson, Technical Mathematics, 3rd edition 5 6.5 Determinants Evaluating Determinants on a Calculator Many of today s scientific calculators allow you to evaluate a determinant quickly (and accurately). The following procedure describes how this is done on a Texas Instruments TI-89 graphics calculator. To evaluate a determinant you will need to use the matrix features of this calculator. We will learn more about a matrix in Chapter 17. Example 6.33 Use a graphing calculator to evaluate the determinant is Example 6.32: 2 1 5 2 1 0 4 5 7 2 1 0 5 0 3 2 FIGURE 6.22a FIGURE 6.22b Solution Begin by pressing the APPS 6 keys to access the Data/Matrix Editor. This will be new matrix so press 3. This is going to be a matrix, so on the first line, Type: press 2. Next, you need to name the matrix so press to go to the Variable: box and enter a name for the matrix, for this example we will use a so press the = key and an a will be entered in the box. This will be a 4 4 matrix, which means that it has 4 rows and 4 columns. Press alpha 4 4. Your calculator screen should now look like the one in Figure 6.22a. ENTER ENTER. We are now ready to enter the elements of our determinant. Start with the element in the upper left-hand corner, 2, and press 2 ENTER. Next enter the second element in the top row, 1, by pressing 1 ENTER ; the third element in the top row, 5, by pressing ( ) 5 ENTER ; and so on until all 16 elements have been entered. When you complete a row, the calculator will go to the left-most element in the next row. When you have finished entering the elements, return to the HOME screen by pressing 2nd QUIT. Now, you are ready to evaluate this determinant. To evaluate a determinant, you want the MATH menu for matrices. To get this, press 2nd MATH 4. The result is shown in Figure 6.22c. There are many matrix operations listed. The one we want, determinant (or det ), is listed second, so press 2. If you named your matrix a, press alpha A ) ENTER. The result, shown in Figure 6.22d, shows that the value of the determinant of [B] is 230.
Peterson, Technical Mathematics, 3rd edition 6 FIGURE 6.22c FIGURE 6.22d