Contents 10. Graphs of Trigonometric Functions

Transcription

1 Contents 10. Graphs of Trigonometric Functions Sine and Cosine Curves: Horizontal and Vertical Displacement Example Composite Sine and Cosine Curves Calculator Graphics Example Example Example Graphs of the Other Trigonometric Functions Example Example Parametric Equations Using a Calculator to Graph Parametric Equations Example Example Example

2 Peterson, Technical Mathematics, 3rd edition Sine and Cosine Curves: Horizontal and Vertical Displacement Example The data in the Table 10.1 gives the number of hours between sunrise and sunset in Seattle, Washington on the 1st and 16th of each month for Table 10.1: Length of Day in Seattle, WA, 2002 Month January February March April Date Hours Month May June July August Date Hours Month September October November December Date Hours Determine a sinusoidal regression for this data. Solution Before we enter the data in a calculator we need to decide on a numbering system for the x-axis. One system would be to let January 1 = 1, January 16 = 16, February 1 = 32, February 28 = 59,..., December 31 = 365. Using this procedure we get the data in Table Table 10.2: Length of Day in Seattle, WA, 2002 Month January February March April Date Day Hours Month May June July August Date Day Hours Month September October November December Date Day Hours The data is entered into the spreadsheet in the same format as the table. (See Figure 10.17a.) 1 Source:

3 Peterson, Technical Mathematics, 3rd edition 3 To graph the data, we must carefully select the appropriate rows that will make up the source data for the series. The rows are Day and Hours. Follow the steps shown below in Figure 10.17b 10.17i to create the scatterplot of the data. FIGURE 10.17a FIGURE 10.17b Click on Chart Wizard and select XY (Scatter). FIGURE 10.17c Click on Next and then select the Series tab. FIGURE 10.17d Place the cursor in the X-Values dialogue box. Then, while holding down the Ctrl key, click and drag to select the cells in each Day row. FIGURE 10.17e Repeat the process for the y-values. Place the cursor in the Y-Values dialogue box. Hold the Ctrl key down and click and drag over the cells in the Hours rows FIGURE 10.17f Complete the process to obtain a graph that is similar to this one. The scale on the y-axis has been adjusted to fit the data. Unlike many other functions, Excel does not have a built-in modeling tool for sine functions. However, a model can be found using SOLVER. The model we will attempt to find is in the form y = asin(bx + c) + d where a, b, c, and d are constants. Construct a template similar to the one shown in Figure 10.17g. Column A and Column B should contain the points from the information given. Column C will contain the model using the four constants in cells D4, E4, F4, and G4: =$D$4*SIN($ E$4*A4F 4)+G4+. Column H will be the squared error between the y-value in Column B and the Model in Column C: =(B4-C4)ˆ2. Column I will contain the sum of the squared errors: =SUM(H4:H27). (Note: it is helpful to have something in Cells D4, E4, F4, and G4.)

4 Peterson, Technical Mathematics, 3rd edition 4 FIGURE 10.17g The idea is to make the Sum of the Squared Errors the very smallest number possible. You can experiment with different values for a, b, c, and d to see what happens to that sum (see Figure 10.17h). FIGURE 10.17h Now, use Solver. Place the cursor on cell I4. Under Tools find Solver. We want to minimize the value in I4 by changing the values in D4, E4, F4, and G4. (See Figure 10.17i.) Solver can produce errors if the values that are going to change are too far from the solution you desire. In this case, the data is sinusoidal with a period of 360 and an amplitude of about 4.5. The curve is shifted vertically by about 12. So, 2?/b = 360 which means b FIGURE 10.17i After several attempts, with different combinations of values for a, b, c, and d, Solver finds that an appropriate model (see Figure 10.17j). (A model is appropriate if the scatter plot of the data and the model-data is about the same. (See Figure 10.17k.)

5 Peterson, Technical Mathematics, 3rd edition 5 FIGURE 10.17j FIGURE 10.17k

6 Peterson, Technical Mathematics, 3rd edition Composite Sine and Cosine Curves FIGURE 10.25a Example FIGURE 10.25b Calculator Graphics Graphing these curves by hand takes a great deal of time. Using a calculator or computer to draw graphs of trigonometric curves would save time. We will briefly describe how to use a spreadsheet to graph trigonometric functions. All of the directions and illustrations are specific to Excel. Use a spreadsheet to graph y = 1 2 cos(3x + π). Solution The first step in graphing any equation is to make a table of values. Notice that the input values will be radian values, not degrees. This is the default setting for trigonometric functions in Excel. The first decision to make concerns the x-values. What values do we use? Your answer may not be exactly what you want the first time, but if it is close you will be able to adjust the scale on the x-axis after the first sketch. We know the amplitude of this function is 1 2 = 0.5 and that it has a period of 2π Since the period is in 1 3 π, we should scale the axis in 1 1 6π, or 12 π, depending on how many values we wish to use. Look at Figure 10.25a. The first column will be the multiples of π 12. The second column will be the x-values; obtained by multiplying π 12 by the number in Column A. This is a way to make filling in the x-values easier and quicker. Next enter the function in cell C2 (see Figure 10.25b): =0.5*COS(3*B2PI())+ FIGURE 10.25c Copy the two formulas in B2 and C2 down to match the values you placed in Column A. (See Figures 10.25c d.) Use the Chart Wizard to construct your first look at the scatterplot. (Use the connected option of the XY-Scatter.) Figure 10.25e shows the graph using the data shown in Figure 10.25d. FIGURE 10.25e FIGURE 10.25d However, if we want to show the graph beginning on the left side of the y-axis, then we can adjust the values in Column A as shown in Figure 10.25f. After adding a few more values in Column A and adjusting the Source Data, the completed graph is shown in Figure 10.25g.

7 Peterson, Technical Mathematics, 3rd edition 7 FIGURE 10.25f FIGURE 10.25g

8 Peterson, Technical Mathematics, 3rd edition 8 FIGURE 10.26a ( ) sin x Example Use a graphing calculator to graph y = (sin 2x). x Solution This is the same function we graphed in Example We will graph this function as the product of two functions: y 1 = sin 2x and y 2 = sin x. Their product x is y = y 1 y 2. Based on our work in Example 10.18, we choose x-values between 0 and 10 scaled by π 4. As we ve done before, Column A will be the multiplier and Column B will contain the multiples of π 4 (see Figure 10.26a). Next, enter the functions. In C2, enter =sin(2*b2), enter =(sin(b2))/b2 in D2, and =C2*D2 in E2. (See Figures 10.26b d.) Copy these values to complete the table as shown in Figure 10.26e. FIGURE 10.26b FIGURE 10.26c FIGURE 10.26d The graph of the function is made using the source data from Column B (the x-values) and Column E (the y-values). The result is shown in Figure 10.26f. FIGURE 10.26e FIGURE 10.26f Example Use a spreadsheet to graph two cycles of y = 3 cos 2x sin 4x + 5. Solution We need to analyze this function to help determine the period and the range. To help our analysis we let y = y 1 + y 2 with y 1 = 3 cos 2x and y 2 = sin 4x + 5. The function y 1 = 3 cos 2x has amplitude 3, period π, and vertical displacement 0. the function y 2 = sin 4x + 5 has amplitude 1, period π 2, and vertical

9 Peterson, Technical Mathematics, 3rd edition 9 displacement of 5. Because the least common multiple of π and π 2 is π, the period for y = y 1 + y 2 is π. The amplitudes are 3 and 1, so the largest possible amplitude is 4. With a vertical displacement of 5, the range of this function is in the interval [5 4, 5 + 4] = [1, 9]. Note that this is probably not the actual range of the function. This range is just an estimate to help establish the values of the y-axis. Since the period is π, an interval from x = 0 to x = 2π will include two cycles. The x-values will go from 0 to 2π by π 8 as shown in Figure 10.27a. Next enter the functions as shown in Figures 10.27b d. FIGURE 10.27a FIGURE 10.27b FIGURE 10.27c Copy the formulas and graph the result. All three graphs are shown in Figure Figure 10.27e and the graph of only the function is shown in Figure 10.27f. FIGURE 10.27d FIGURE 10.27e FIGURE 10.27f

10 Peterson, Technical Mathematics, 3rd edition Graphs of the Other Trigonometric Functions Calculator Graphics Example Use a spreadsheet to graph of y = 1 3 tan ( 2x π 4 ). Solution From Table 10.5, we see that the period is π We would like to see at least two periods of this graph, so we will want to use values of x between π 4 and 9π 8, between 0.8 and 3.6. Spreadsheets aren t built to handle asymptotes. We expect to see asymptotes when we graph the tangent function, so we should know up front that we will have to use a great many more points than usual to construct an accurate graph. In the past few examples, we ve used multiples of some fraction of π to make the table of values. In this case of the tangent function (and other functions that have asymptotes), it actually works out best not to put in the exact values. Figure 10.34a shows a graph of this function without connecting the points and using x-values incremented by 0.1 between 0.8 and 3.6. By making the increment for x smaller, and adding more points, the graph can be more accurate (see Figure 10.34b). Part of the table of values used for Figure 10.34b is shown in Figure 10.34c. FIGURE 10.34a FIGURE 10.34b By connecting the dots a more recognizable representation of the graph of the function is created, as shown in Figure 10.34d. FIGURE 10.34c FIGURE 10.34d

11 Peterson, Technical Mathematics, 3rd edition 11 Notice that the computer seems to have drawn in the vertical asymptotes. Look closely. These are not vertical lines. The portion above the x-axis is not directly above the portion below the x-axis. These apparent vertical asymptotes are put in by some graphing calculators and computer graphing programs as they try to connect the last point on the left side of an asymptote (a large negative number) to the first point on the right side of the asymptote (a large positive number). Example Use a spreadsheet to sketch y = 1.5 sec (x + 2). Solution The x-values are between x = 1 and x = 7. Since we expect asymptotes in the graph of this function, we will set up the spreadsheet to allow us to change the increment for x until we are satisfied with the result. The secant is not a built-in function so we will need to use the reciprocal identity sec x = 1 to rewrite this function as cos x y = 1.5 cos(x + 2) The first sketch is shown in Figure 10.35a. The graph is not as smooth as expected. Figure 10.35b shows the result after changing the increment to 0.1. Once again, you see that the computer seems to have drawn in the vertical asymptotes. FIGURE 10.35a FIGURE 10.35b

12 Peterson, Technical Mathematics, 3rd edition Parametric Equations Using a Calculator to Graph Parametric Equations Example Use a spreadsheet to sketch the curve represented by the parametric equations x = 2t and y = t 2 4. Solution There is nothing difficult about graphing parametric equations using a spreadsheet. In fact, the spreadsheet makes it easy. Create 3 columns, one for t, one for x, and one for y. (See Figure 10.50a.) Next enter the values for t from 4 to 4 by one, enter the equation for x in cell B2 (see Figure 10.50b), and enter the equation for y in cell C2 as shown in Figure 10.50c). Copy the two equations and graph the result (see Figure 10.50d) using Column B and Column C as your source data. FIGURE 10.50a FIGURE 10.50b FIGURE 10.50c FIGURE 10.50d

13 Peterson, Technical Mathematics, 3rd edition 13 Example Use a spreadsheet to sketch the curve represented by the parametric equations x = 2 cos t and y = sin t. Solution We choose t values between 0 and 6.3 in increments of 0.1. You should type =2*cos(A2) in cell B2 and =sin(a2) in cell C2. The resulting table of values is shown in Figure 10.51a and the resulting graph is displayed in Figure 10.51b. FIGURE 10.51a FIGURE 10.51b Gravity-influenced Trajectories When a communications satellite or the Space Shuttle is launched, it is given enough velocity to enable it to go into Earth s orbit instead of falling back to Earth. How much velocity is enough? The answer depends on the weight of the rocket and where you want the orbit to be. The Space Shuttle, for example, must be sent off at 17,500 mph in order to reach its orbit. If a projectile moves with an initial velocity that is below its escape velocity, it returns to Earth without going into orbit. We ll now return to motion which does not reach orbit, and is therefore called sub-orbital motion. Summary of Sub-Orbital Projectile Motion The parametric equations that describe the position of an object thrown into the air with velocity less than the Earth s escape velocity are { x(t) = vx0 t + x 0 y(t) = 1 2 gt2 + v y0 t + y 0 where t is the time after the toss; x(t) the horizontal distance traveled; y(t) the vertical distance traveled; v x0 the initial horizontal component of velocity (v x0 is constant during flight); x 0 the horizontal distance from zero point at release; g = 32 if the units are feet and seconds and g = 9.81 if the units are meters and seconds; v y0 is the initial vertical component of velocity; and y 0 is the height above ground at release. Example A ball is thrown upward at an angle of 50 from a height of 30 ft with a velocity of 92 ft/sec. (a) Write the parametric equations of the position of the ball at any time t. (b) Plot the trajectory on your calculator.

14 Peterson, Technical Mathematics, 3rd edition 14 (c) Estimate to two decimal places the time when the ball reaches the ground. (d) Estimate to the nearest foot the maximum height the ball reaches. (e) Estimate to the nearest foot the horizontal distance the ball traveled. Solutions: We are given that y 0 = 30, and we can set x 0 = 0. Since v 0 = 92, we have v x0 = 92 cos ft./sec., and v y0 = 92 sin ft./sec. Since y 0 and v 0 are both in terms of feet, we use g = 32. (a) Using the above information, the equations are { x = 59t y = 16t t + 30 (b) The trajectory is shown in Figure 10.52a along with the table of values used to create the trajectory. (c) There are several ways to answer part (c). Using the quadratic formula to solve 16t t + 30 = 0, we see that t 0.39 seconds or t 4.77ṡ. The first time is unacceptable since it was before the ball was thrown. Thus, it took about 4.77 seconds for the ball to hit the ground. The table of values provides another way to estimate the time. In Figure 10.52a, we see that the y-value becomes zero sometime between t = 4.7 s and t = 4.8 s. A more exact answer can be found by choosing smaller increments of t (see Figure 10.52b). FIGURE 10.52a FIGURE 10.52c FIGURE 10.52b A third way is to use Goal Seek. Figure 10.52c shows Goal Seek being used to set cell C61 to a value of 0 by changing cell A61. The result is shown in Figure 10.52d. The projectile will hit the ground in seconds. (d) In the last chapter, we learned that the vertex of a parabola f(t) = at 2 + bt + c occurs when t = b 2a. Here we have y(t) = 16t2 + 70t + 30, so the vertex is at t = 70 2( 16) = By using t = , a height of about feet will be obtained (see Figure 10.52e).

15 Peterson, Technical Mathematics, 3rd edition 15 FIGURE 10.52d FIGURE 10.52e You can also use Solver to find the maximum value of cell C14 by changing the value of cell A14 (see Figure 10.52f). The result is shown in Figure 10.52g. FIGURE 10.52f FIGURE 10.52g (e) At the time the ball hits the ground (when t = 4.77 seconds), the ball has traveled x(4.77) feet horizontally. Since x(4.77) = , the ball lands feet away from the point where it was thrown.