Download Using Your Calculator

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