3. Trigonometry on the Calculator.

We note that the calculator has keys labeled SIN, COS, and TAN. These are, as might be expected, the keys for the sine, the cosine, and the tangent functions respectively. If the real number x is on level 1 of the stack, pressing SIN will give sin(x), TAN will give tan(x), and COS will give cos(x). The real number x will be interpreted as degrees or radians according to the angle mode setting. For example if the number 30 is on level 1 of the stack and the calculator is in degree mode, pressing SIN gives the result .5 as expected. If, however, the calculator is in radian mode with 30 on level 1 of the stack, the result of pressing SIN is -.988031624093, which is the sine of 30 radians.

We note that there are no keys for the cosecant, secant, or cotangent functions. Since these functions are the reciprocals of the sine, cosine, and tangent respectively, they can easily be obtained with the keys we have followed by the 1/x key. For information on the trigonometric functions see page 10-8, 12-2, and A-2 of UG.

Calculator Example 3.3.1

Find 

Solution: With the calculator in radian mode key in LS  COS 1/x. We see the result 2.00000000001. The answer, of course, should be 2, but we are again seeing an example of the inevitable round off errors caused by using a finite machine to approximate computations with real numbers.

We see that the left shift functions for SIN, COS, and TAN are labeled ASIN, ACOS, and ATAN, respectively. These are, respectively, the Arcsine, Arccosine, and Arctangent functions. These functions are not inverses of each other since, for example, SIN is the sine function, not the Sine function.

Calculator Example 3.3.2

Find  then take the Arctangent of the result.

Solution: With the calculator in radian mode key in 3 LS  TAN, and we see the expected answer -1. Now key in LS ATAN and we get the result -.785398163397, which is the decimal approximation for  This should not be surprising since  is not in the domain of Tan and so  is not necessarily  for this angle.

Before starting the the next example, you may want to review the plotting instruction in UG, particularly pages 22-1 to 22-2 and 23-1 to 23-4.  Calculator Example 3.3.3 Graph y = sin(x) foron the calculator.  Solution: The first step is to insure your calculator will react as indicated by these instructions. If you have variables called X, EQ, and/or PPAR in your variable list, purge them. (See page 5-10 of UG.) Set the calculator to degree mode. Now key in RS PLOT to get into the PLOT dialog box. Press ' SIN m-X ENTER to enter 'SIN(X)' into the EQ: box. Now key in RA 180 +/- ENTER 360 ENTER to set the H-VIEW: boxes. Now press m-CHK to set the V-VIEW: box to AUTO. The dialog box has been set, (see Figure 1a). Now press m-ERASE m-DRAW to draw the graph which we see in Figure 1b.

Figure 1a Figure 1b

The tick marks on this graph are still set to the default of 10 pixels. We would like them to be set to 45 on the horizontal axis and .25 on the vertical axis. Press CANCEL m-OPTS to get into the options dialog box. Use DA to get to H-TICK: and change it to 45, change V-TICK: to .25, then press m-CHK to unselect _PIXELS. The options dialog box should now look like Figure 1c. Press m-OK, then m-ERASE and m_DRAW to get the graph as in Figure 1d. Notice that the tick marks now make more sense.
 

Figure 1c

Figure 1d

Calculator Example 3.3.4

Graph y = Cos(x) and y = Arccos(x) on the same axes. See Problem 7, Exercises for Chapter III Section 2.

Solution: Assuming your calculator is as you left it from the previous example, press CANCEL or m-CANCL to return to the PLOT dialog box. Use the arrow keys to get to the angle mode and set it to radians. Go to the EQ: box and put 'COS(X)' in it. Now go to the first field in H-VIEW: and press NXT m-RESET DA m-OK. This resets the plot parameters back to their default state. In this state one unit in both directions is 10 pixels, so geometric properties will not be distorted, but the two views are about twice as big as we need them. We will cut each of the fields in the two view areas in half, and will add .75 to the two V-VIEW fields to move the origin down that amount. To accomplish this, press m-CALC 2  m-OK. Now move to the second field in
 

H- VIEW: and repeat. For the two V-VIEW: fields you will insert .75 + after the division. Now press NXT m-OPTS. Set the value of LO: to 0 and move to HI:. We want this value to be but we can't put in the symbol, we must enter a decimal approximation. Press NXT m-CALC DROP LS  m-OK. Press m-OK again to get back to the PLOT dialog box then m-ERASE m-DRAW to produce the graph in Figure 2a. Pressing the minus sign will remove the menu so you can see the whole graph. Pressing the minus sign again brings the menu back. Return to the PLOT dialog box and change EQ: to 'ACOS(X)' and in the OPTIONS dialog box change the domain, that is LO: and HI: to -1 and 1 respectively. Now return to the PLOT dialog box and press m-DRAW. NOTE: Do not press ERASE in this case since we want this graph to be superimposed on the previous one.  See Figure 2b.  Finally, to put in the line of symmetry, press m-(X,Y) to show the coordinates at the bottom of the screen and use the arrow keys to move the cursor to X:2 Y:2. Now press m-EDIT NXT m-MARK, then use the arrow keys to move the cursor to the origin. Now press NXT NXT m-LINE to draw the line, and m-MARK to remove the last mark. You should now have the graph as in Figure 2c.

Figure 2a  Figure 2b  Figure 2c