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Using Your Calculator A Guide for First Year Engineering Students This booklet is a step-by-step guide which enables you to use your scientific calculator skillfully and creatively. It assumes that you are using the Casio fx-991MS calculator. Much of what is said also applies to the fx-991 ES calculator, too. The advantage of using 991 ES is its Maths display. To make it on use shift mod e 1 , where you can type the terms as you write on paper. It was not possible to cover all possible calculator functions in this small booklet. Still, an attempt has been made to explain many of the functions which are commonly used by engineering students. A fairly large number of graded questions has been included. These exercises cover a number of topics, such as basic calculations and hierarchy of operations, use of parentheses, the use of mental arithmetic, trigonometric functions, integration, solution of quadratic and simultaneous equations and the more exotic uses such as solution of non-linear algebraic equations by iterative methods. The recommended method to use this booklet is to go through every problem sequentially, at the same time trying it out on your calculator until you reach the right answer. At the appropriate places, comments have been made which should be understood well. The student should by all means add his own remarks, wherever needed. It is hoped that by the time he reaches the end of the exercises, the student will have developed sufficient skill in using his calculator. As far as possible, whenever a key is mentioned by name, it is put in a rectangular box (e.g. press Ans ). However if a number is to be entered, then the number is shown as it is, i.e. without a box. (e.g. 4 instead of 4 ). Also, some functions get activated only after first pressing the Shift key or sometimes the Alpha key. In such cases, this prior key-stroke is not explicitly mentioned. e.g. “Press sin 1 ” would actually mean “press Shift , followed by sin 1 ”. In some cases the final key which has to be pressed at the end of many operations may not be explicitly mentioned. The information contained in this booklet is provided ‘as is’ without warranty of any kind, express or implied, including without limitation any warranty concerning the accuracy, adequacy, or completeness of such information or the results to be obtained from using such information. Vidyalankar Classes shall not be responsible for any claims attributable to errors, omissions, or other inaccuracies in the information contained in this booklet, and in no event shall Vidyalankar Classes be liable for direct, indirect, special, incidental, or consequential damages arising out of the use of such information. 1 Evaluate : 1. 5 3 + 2. Would you expect the answer to be 17 or 25 ? Why ? [Ans.: 17] Conclusion : has higher priority than +. 2. (5 3) + 2. [Ans.: 17] Since has higher priority, the brackets are unnecessary i.e. the answer is the same regardless of the presence or absence of the brackets. 3. 5 ( 3 + 2 ) = 25. The answer now changes, because the bracket has highest priority. 4. 4 2 2 = 1. Here both signs have equal priority. Hence the expression is evaluated from left to right i.e. (4 2) 2 = 1. This brings us to the hierarchy of operations. The usual sequence, in decreasing order of importance is 1st priority 2nd priority 3rd priority 4th priority 5th priority 6th priority : : : : : : ( ) brackets xy exponentiation functions like sin x, cos x, log x etc. , , i.e. multiplication, division. , , i.e. addition, subtraction. in case two operations have the same priority, then the evaluation is done from left to right. 5. 3 52 = 75. Note that the exponentiation (i.e. squaring) takes place before multiplication. 6. (3 5 )2 = 152 = 225. Bracket has highest priority. 7. 4 2 10 = 20 8. ( 4 2 ) 10 = 20. 9. 4 (2 10 ) = 0.2 Note that in (7), and have equal priority. Hence the expression is evaluated from left to right. Hence answers of (7) and (8) match. But (9) is different. 1 . Which way will you proceed? 1 22 or 1 ( 22 ) ? 2 2 Does the answer match in both methods ? 10. We want to find 5 = 5 2 5 ? or is it = 5 ( 2 5 ) ? or = 5 2 5 ? 2 5 Note that the first method yields a wrong answer. Observe that in the 3rd method (which is the most commonly used), two ‘’ signs have been used, although the original question contains one ‘’ and one ‘’ sign. 11. Is 12. 10 = ? Are brackets necessary, anywhere ? 22 2 13. 2 3 sin 30 = 4 1 in fx -991 MS and 2 3 sin (30) = in fx-991 ES. 3 3 Since in ES calculator consider this as 2 3 sin (30). So it is better to use Math form i.e. 2 . 3sin 30 2 9.8 10 = 14. Note that entering 14. wrong answer, as the 2 9.8 10 gives you the sign is then applicable only to 2. Hence insert the appropriate brackets for the entire expression. i.e. (2 9.8 10) . 15. (a) 22 4 10 = 40. (Where do you close the brackets ?) (b) Find v if v3 = 64s2, when s = 27. [First find 64 272. Then find 3 Ans. . The final solution should be v = 36]. 16. (a) 1 2 102 3 5 2 = 0.78125. Enter this problem AS IT IS. After you get the right answer, try some mental simplification before entering the numbers.(You can type the expression as 100 2 64 ) (b) Find 43/2. [Ans.: 8 ] Hint: 4 ^ (3 2 ). ^ = “raised to” in fx-991 MS. 17. (a) cos245 = 1/2. Note that entering as it is produces a syntax error; the correct way is to interpret it as (cos 45)2. Also note that if the mode is set to ‘degrees’, you don’t have to tell the calculator the units of 45. Otherwise, use the DRG entering the number 45. key appropriately, after (b) tan2 60 = 3. (c) cot2 30 = 3. 18. sin 45 × 2 = 1.414, without brackets = 1, with 45 × 2 in brackets. (why ?) 19. 20. 9.8 9 2 121 cos 45 9.8 9 = 0.515 = 0.515 c 2 121 cos 4 without converting to degrees. Use the [Try keying in 4 3 DRG Function]. 21. In Q.19, try to edit the above expression by using the and keys to position the cursor on the c 1st line of the display, and then using the DEL key to delete say . Now replace it by, 60 4 (say). Then position the cursor on the very first character of the expression, and delete 9.8. Now go into insert mode by the INS key and insert the number 10 at the beginning. Press INS again, to turn ‘insert’ off. Press to get the new answer. Note that DEL acts similar to the DEL key in a personal computer. HOWEVER, when the cursor is one position to right of the ENTIRE keyed expression, DEL will act like the computer’s backspace key. 22. 9.8 3.12 . 2 11.22 cos 2 45 23. Conversion between rectangular and polar co-ordinates : We know that the polar co-ordinates (r = 10, = 300) correspond to the rectangular co-ordinates (x = 5 3 , y = 5) by using the formulae : x = r cos and y = r sin However it is possible to convert directly from one co-ordinate system to another on your calculator. Polar to rectangular : Type Rec( , followed by 10, and then a comma, and then 30. Finally, close the bracket. Make sure that you have entered the angle in the proper units (in this case, degrees). Finally press . The respective values of x and y i.e. 8.66 and 5 get assigned automatically to the respective variables E and F, which can be recalled in the normal way. In 991 ES you get directly x = 8.66 and y = 5. Rectangular to polar : Type Pol( , followed by5 3 , then a comma, and then 5, followed by the closing bracket. Press . The values of r (i.e. 10) and (i.e.30/0.5236C etc. depending on the mode you have selected) automatically get stored as the variables E and F respectively; they can be recalled in the normal way. In 991 ES you get directly r = 10, = 30. gx 2 24. Consider the formula y = x tan – 2 , used in projectiles. 2u cos 2 Find y, if x = 11.3, = 30°, g = 9.8, u = 12. [Ans.: 0.7307] Now put = –30°, (don’t simply edit, but retype the entire expression) and prove that y = –12.32. 25. Find an acute , which satisfies the trigonometric eqn. 429.3 cos – 500 sin = 5. Note : In both 991 MS and 991 ES, this question can be easily solved using the SOLVE key of your calculator. In that approach, one has to type the equation, which is then solved by the calculator automatically using Newton’s method. The answer is an approximation. 4 However, it is recommended that the entire, analytical solution is presented, because some marks are reserved for the steps. Hence, the preferred method is : 429.3 cos – 500 sin = 5 Divide by 429.32 5002 (Do this on your calculator). 0.6514 cos – 0.7587 sin = 7.587 10–3 (check!) Now find sin–1 (0.6514). This will be the same as cos–1 (0.7587) (why?), and equals 40.65°. Also sin–1 (7.587 x 10-3) = 0.4347 sin (40.65°) . cos – cos (40.65°) .sin = sin (0.4347) using formula for sin (A - B), sin (40.65 – ) = sin (0.4347) Comparing, 40.65 – = 0.4347 = 40.22°. Do this calculation yourself, and see if you get the final answer. Note : = 40.22° is an acute angle and hence satisfies the requirements of the problem. If not, then other solutions have to be found by using formula such as sin (180 – ) = sin , sin (360 + ) = sin etc. 26. Find an acute angle , satisfying 4 cos – 3 sin = 2. [Ans. : = 29.55°] 27. Find a positive (acute or obtuse), satisfying 4 cos + 3 sin = 2. Hint : you will eventually get sin( + 53.13°) = sin (23.58°) Now + 53.13° = 23.58° gives = –29.55°, which is not a positive . Remember, DO NOT then assume that = +29.55° is the correct answer. It can be verified to be incorrect by substituting in the original equation. using sin = sin (180 – ) on the RHS, we obtain sin ( + 53.13°) = sin (180 – 23.58) = sin (156.42°) Now, + 53.13° = 156.42° 103.29°, the required positive angle. Note here that if you are trying to solve 4cos + 3sin = 2 by the SOLVE key, then you are at an obvious disadvantage here. The equation has more than 1 root, as shown above, and which root you get depends on the initial approximate value which you choose for . However chances of getting correct answer are more if you choose approximate value logically. 28. Consider the formula R = u 2 sin 2 . Find that positive, acute angle , which corresponds to g R = 35.35, u = 20, g = 9.8. 5 gR = 0.8661. Now sin –1 (0.8661) = 60° 2 u sin (2) = sin (60°) = sin (120°) (why?) 2 = 60° or 120° = 30° or 60° There are 2 positive acute angles possible. Solution : sin (2) = u 2 sin 2 29. (a) Consider the formula R = . g corresponding to R = 40, u = 30 and g = 9.8. Find those positive, acute values of , [Ans. : = 12.9 or 77.1] [Ans. : = 9.533 or 80.467] (b) What if R = 30, u = 30, g = 9.8 ? 30. We know that cos x dx sin x 0 . 0 To do this on your calculator, use the dx key and then input cos xc. Remember the units; because cos x dx cos x c dx . This is the most likely place where one can go wrong ! Now put a comma, and then type the lower limit 0, followed by a comma, and then the upper limit . Now put another comma, followed by the number 4. Lastly type the right bracket and then press . The last number ‘4’ corresponds to the number of intervals, which has to be specified during the numerical integration method which the calculator uses. Pressing the key gives you the correct answer, 0. In 991 ES you type the limits as you write on paper dx . Note that the number of intervals can be any integer in the range 1-9. Even if you skip this step, the calculation will still occur, which may mean that the calculator chooses some value which it feels suitable. Also note that the recommended method of definite integration is to actually show all the steps, as in std XII. Thus cos xdx sin x 0 sin sin 0 0. Remember, marks may be cut if you write the 0 answer directly, using method of using the dx dx key of your calculator. However, we have shown you the key, just as a matter of interest. 4 t 200 200 31. Find cos dt 2 0 [Ans. : 254.6] (If you get an incorrect answer, check if the angle has been entered in radians.) 6 0.5 32. Verify that 9e 6 t dt 1.425 . Note that is some calculators the exponent of e is displayed 0 in the same line, and may confuse you. Thus, e2 may be actually displayed as e2. 33. Simultaneous equations can also be solved directly on the 991 MS and 991 ES both. However, whenever you have to solve simultaneous equations the examiner requires that you write all the steps and then only the answer. There are marks reserved for these steps. Do not find the answer directly on calculator and proceed to write this answer! However, the method is discussed below, again only as a matter of interest. Solve the 3 simultaneous equations : –0.5x + 0.664y = 0 –0.866x – 0.5y - z = 0 –0.557y = 4000 [Ans. : x = –9,537, y = –7,181, z = 11,850] Solution : Change the mode of the calculator to Equation Mode by repeatedly pressing the mode key until the EQN appears. Then select the number corresponding EQN. Select the number of unknowns as 3. Now compare the given equations with a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3 Key in the value of a1 and confirm it by pressing Do the same for all the other constants. key or After the final is pressed, the values of x, y, z can be seen on screen, using the the key. Note: For 991 ES mod e 5 2 34. Solve: 0.408x – 0.535y + 0.1374 z = 0 –0.408x + 0.267y + 0.549 z = 0 0.816x + 0.802y + 0.824 z = 150 [Ans. : x = 84.8, y = 71.9, z = 28.07] 35. Solve the quadratic equation x2 – 4x + 3 = 0. Solution : Here again, the examiner would require that you write the steps. So it is not advised to solve a quadratic by calculator. But the method is given, again as a matter of interest. Press mode repeatedly until you see EQN in the displayed options. Select EQN. Scroll to the right by pressing and then specify the degree as 2 [if it is a cubic equation, then specify 3]. Note that if you are trying to solve simultaneous equations, you must specify the number of unknowns. However, for quadratic/cubic equations, specify the degree. Now compare the quadratic equation with ax2 + bx + c = 0. Key in the value of ‘a’ as 1, followed by . Repeat the procedure for b and c, keying in –4 and 3 respectively. Note : For 991 ES mod e 5 3 The calculator displays x1 = 3 (1st root). Pressing or gives you x2 = 1. To clear the EQN mode, press CLR and select “All”, and then twice press: . 7 36. Solve : 2x2 + 3x – 8 = 0 [Ans. : x1 = 1.386, x2 = –2.886] 37. Solve : x2 + 8x+ 25 = 0 [Ans. : x1= –4 + 3i, x2 = –4 – 3i] 38. Solve 2x2 + 3x + 8 = 0 [Ans. : –0.75 1.854i] 39. Cubic equations can be solved similarly. Can you prove that the real root of 2t3 + 5t2 = 99 is t = 3, while the 2 complex roots are –2.75 2.99i ? Note that the number 99 has to be first taken to the LHS and then entered as 99. 40. One last point : be careful when using the Ans key in memory storage operations. The Ans key, when used in conjunction with memory storage, consider that you want to find 32 42 and then store it in M. Suppose you first key in 32 + 42 . The number 25 is displayed. Now do Ans . The number 5 is displayed, which now becomes the current value of the “Ans” variable. On the 1st line of display, you still have Ans . If you now press STO M , not 5, but 2.223 i.e. 5 gets stored as M! This is because the calculator uses the latest value of the “Ans” variable. To store 5, the proper sequence showed have been: 32 + 42 Ans STO M 8 Engg/Using your Calculator