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Getting started
Texas Instruments TI-85 and TI-86 calculators
Overview: Your graphing calculator or computer is a powerful and flexible tool, which you would
probably be able to use fairly well without reading any instructions. It is important, however, to learn
how to take advantage of some of its not-so-obvious features and how to avoid making errors using it.
Study these instructions and be sure you can work the tune-up exercises at the end.
Topics:
• Basic operations
• Priority of operations in calculations
• The dangers of using improper parentheses
• Exact and approximate decimal values of functions
Basic operations
Press the ON key to start the calculator. Press 2nd followed by the up cursor key N to increase
display contrast and by H to decrease it. Change the four AAA batteries as soon as the screen dims
when graphs are generated. Press 2nd MODE . The screen should show
Normal Sci Eng Func Pol Param DifEq
Float 012345678901 Dec Bin Oct Hex
Radian Degree RectV CylV SphereV
RectC PolarC dxDer1 dxNDer
The words printed in bold type here should be highlighted on the screen. If another item is highlighted or
you want to change a selection, use the cursor keys to move the flashing box to the correct item and press
ENTER . <Normal> denotes normal notation for decimals; <Sci> is for scientific notation; and <Eng>
for engineering notation. With <Float> selected decimals are printed with twelve digits. Choosing an
integer instead of <Float> causes that many digits be shown after decimal points. (Use the second 0
for ten digits and the second 1 for eleven digits.) <Radian> is for radians and <Degree> for degrees.
<Func> is selected to generate graphs y = f (x) of functions. <Pol> is for p olar coordinates, <Par> is
used with parametric equations, and <DifEq> is for differential equations. The other selections will be
explained as needed.
Press EXIT or 2nd QUIT to return to the home sc reen and then GRAPH for the first row of the
graph menu. Press MORE to see the second row of the menu and then press F3 for <FORMT>. The
screen should read
RectGC PolarGC GridOff GridOn
CoordOn CoordOff AxesOn Ax esOff
DrawLine DrawDot LabelOff LabelOn
SeqG SimulG
with the words in bold highlighted. With <RecGC> and <CoordOn> selected, rectangular coordinates
are used and the coordinates of the cursor are displayed with graphs. Points on graphs are connected
if <DrawLine> is chosen and not with <DrawDot>. Use <SeqG> (sequential graphs) to have two
or more graphs drawn one after the other, and < SimulG> (simultaneous graphs) to have them drawn
at the same time. If < GridOn> were selected, dots would be placed on the screen at the points whose
coordinates correspond to the tickmarks on t he axes. The axes would not be shown with <AxesOff>
and labe ls would be displayed with <LabelOn>.
The key 2nd activates the yellow commands above the keys. EXIT is used to return t o a previous
screen and to remove menus. 2nd QUIT returns you to the home screen where calculations are made.
CLEAR with the cursor on a blank line of the home screen clears the screen. In other cases it clears the
line with the c u rsor or removes a menu.
The key ALPHA puts the calculator in upper-case alpha mode, activating the blue letters and
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Getting started TI-85/86 calculators, 2
other symbols above the keys. Pressing ALPHA ALPHA locks the calculator in upper-case alpha mode
and then pressing ALPHA or ENTER takes it out of up per-case alpha-lock mode. 2nd ALPHA puts
it in lower-case alpha mode. Entering a number followed by
STOI , one or more letters, and ENTER
assigns that number to th e letter or letters. The number can then be recalled by entering the letter or
letters. The calculator is locked in alpha mode after STOI is pressed.
If you make an error in a command or calculation, the type of error is given and a menu appears.
Select <GOTO> to go to the error to correct it or <QUIT> to cancel the incorrect command.
In the home screen, 2nd ENTRY recalls the last expression that was evaluated so it can be edited,
if necessary, and used again. The ON key stops the generation of graphs, the running of programs, and
other operations. The ENTER key can be used to interrupt and resume the generation of graphs.
Refer to the owner’s manual for further information.
Priority of operations
The meaning of a formula involving functions, powers, sums, differences, products, and quotients depends
on how the formula is interpreted to determine the order in which the operations are performed. Texas
Instruments TI-86 calculators in most instances interpret formulas with the following rules, which are
those generally used in manual calculations.
Rule 1 Operations are performed from left to right, except as described in Rules 2 through 5 below.
Rule 2 Expressions inside parentheses are evaluated as soon as they are reached.
Rule 3 Addition and subtraction have the lowest priority. If an addition or subtraction is followed by
multiplication, division, a power, or a function, the addition or subtraction is postponed until another
addition or subtraction or the en d of the expression is reached.
Rule 4 Multiplication and division have medium priority. If a multiplication or division is followed by
a power or a function, the multiplication or division is postponed until the power or function has been
evaluated.
Rule 5 The taking of powers and evaluation of functions have the highest priority and are performed
as soon as they are reached.
Example 1 (a) Calculating 5 + 2√9 involves addition, multiplication, and the taking of a sq uare
root. In what order are these operations perfomed? (b) Find the value of 5+ 2√9 with
your calculator.
Solution (a) By Rule 5 above, finding the square root has the highest priority and is performed
first, yielding 5 + 2√9 = 5 + 2(3). Multiplication has the next priority, by Rule 4, and
gives 5 + 2(3) = 5 + 6. The remaining addition gives 5 + 6 = 11.
(b) Press 5 + 2 2nd
√
9 so the screen reads 5 + 2
√
9. Then press ENTER
for the answer 11.
3(4)
Example 2 (a) What steps would you use to evaluate
√
36
3(4)
√
36
−
10
on your calculator.
8 − 3
Solution (a) Working from left to right, you would first multiply the 3 and 4 to have
−
10
? (b) Find the value of
8 − 3
√
12
36
10
. Then you would evaluate the square root, yielding
8 − 3
6 would give 2 −
denominator to have 2 −
10
. Next, you would perform t h e subtraction in the remaining
8 − 3
10
. Finally, you would divide 5 into 10 to obtain 2 − 2 and
5
12
10
−
6
. Dividing 12 by
8 − 3
subtract for the answer 0.
−
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Getting started TI-85/86 calculators, 3
(b) Press 3 ( 4 ) ÷ 2nd
√
( 3 6 ) − 1 0 ÷ ( 8 − 3 ) so
the screen reads 3(4)/√(36) − 10/(8 − 3). Then press ENTER for the value 0.
The next example shows how using the negation symbol–for subtraction can lead to an error
message or give an incorrect result because a product is calculated instead of a difference.
Example 3 Evaluate the expressions 2π − π, 2π–π, 8 − 5, and 8–5, where − is the subtrac tion
symbol and
–
is the negation symbol. Explain the results.
Solution The calculator gives 3.14159265359.= π for 2π − π, which it obtains by subtracting
π from 2π. It gives −19.7392088022.= −2π2for 2π–π, which it interprets as 2π
multiplied by −π. The expression 8 −5 equals 3, and you get −40 with 8–5, which the
calculator either interprets as the product of 8 and −5.
The need to use ∗
Because TI-85 and TI-86 calculators allow words to be used for variables, multiplication signs (∗) must
be used between letters that rep resent numbers to be multiplied.
Example 4 Evaluate AB with A = 5 and B = 2 by first storing the values of A and B.
Solution Enter
5 STOI A ENTER 2 STOI B ENTER to store the values. Then enter
ALPHA A ∗ ALPHA B ENTER for the answer A × B = 10. (Notice that using
ALPHA A ALPHA B to write AB and then ENTER yields an error message since
the variable AB has not been defined.)
The dangers of using improper parentheses
TI-85 and TI-86 calculators interpret certain expressions in unexpec ted ways because they use the
following modification of Rules 3 through 5.
Rule 6 The taking of powers has priority over the evaluation of functions that appe ar before their
variables, such as the trigonometric functions, logarithms, ex,√, and negation. Also, the parentheses in
expressions such as sin(2) and e ∧ (2) are ignored.
Example 5 Evaluate sin3(2) = (sin(2))3.
Solution The seemingly logical expression sin(2)∧3 will not work. By Rule 6, the parentheses are
ignored, leaving sin 2 ∧ 3. Then the taking of the cube has priority over the evaluation
of the sine function, and t h e calculator gives 0.989358246623.= sin(23) = sin(8).
†
For the correct answer, use an extra pair of parentheses by entering (sin(2)) ∧ 3.
This gives the correct value 0.751826944669.
The TI-85 also uses the following two additional modifications of Rules 1 through 5
Rule 7 Multiplication by juxtaposition has priority over division and multiplication represented by ∗.
Example 6 Attempt to evaluate
1
(10) = 2 by entering 1/5(10).
5
Solution The TI-85 evaluates 1/5(10) as 1/(50) = 0.02 because it uses Rule 7 and multiplies the
10 and the 5 before performing the division. Enter 1/5 ∗ 10 or (1/5)(10) instead.
The TI-86 gives
1
(10) = 2, as expected, because it does not use Rule 7.
5
Rule 8 Multiplication represented by juxtaposition, where the second term is a number or a variable,
has priority over the evaluation of functions that appear before their arguments.
†
If you obtained 0.13917310096 here, then your calculator is using degrees instead of radians. Press 2nd MODE , put the
cursor on <Radian> and press ENTER to select radian mode. Press 2nd QUIT 2nd ENTRY to return to the home screen
and recall the last typed line, and then ENTER for the correct an s wer.
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Getting started TI-85/86 calculators, 4
Example 7 Attempt to evaluate√4π by entering√4π
Solution The TI-85 calculator reads√4π asp(4 ∗ π).= 3.54490770181 because it uses Rule 8
and multiplies the 4 and the π before taking the square root. Enter (√4)π or√4 ∗ π
instead to obtain the correct value 6.28318530718.
The TI-86 yields (√4)π = 2π.= 6.28318530718, as expected, because it does not use
Rule 8.
Example 8 Atempt to evaluate sin(5)(10) by entering this expression in the calculator.
Solution The TI-85 gives the wrong value sin(50).= −0.262374853704 because it uses Rule 8
and does the multiplication before evaluating the sine. Use sin 5 ∗ (10) or (sin 5)(10)
instead.
The TI-86 yields the correct answer 10 sin(5).= −9.58924274663 be cause it doe s not
use Rule 8.
Exact and approximate decimal values of functions
Since some but not all numbers can be represented exactly as finite decimals, it is important to distinguish
exact expressions, such as
need to recognize when coordinates obtained from graphs generated by calculators and computers are
approximations.
Example 4 Use your calculator to complete the table below of ten-digit values of 5x
√
5
exact, but −15.53616253 is only a decimal approximation of 5(−30)
be represented by a finite de cimal. Its value to 20 decimal places, for example, is
−15.53616252976929433439. Which y-values in the completed table in addition to −15
do you recognize as exact?
x y = 5x
1
and π, from decimal approximations, such as 0.33333 and 3.14159. You also
3
3
x at x = −27, −30, 4, 6, 8, and 10. The value 5(−27)
.
1/3
= x y = 5x
1/3
= 5(−3) = −15 is
1/3
, which cannot
.
1/3
=
1/3
=
−27 −15 −30 −15.5361625298
4 10
6 8
Solution You can do these calculations more efficiently by storing the formula for the function.
Press GRAPH F1 to access the y(x) = menu and CLEAR to erase any previous
formula for y1. Press 5 x −V AR ∧ ( 1 ÷ 3 ) to have y1 = 5x ∧ (1/3).
To find the value of the function at x = −27, press 2nd QUIT to return to
the home screen and press (–) 2 7 STOI x−V AR 2nd : 2nd ALPHA Y
ALPHA ALPHA 1 so th e screen reads −27 → x: y1. The colon (above the period
key) separates the two commands on the one line. Then press ENTER for the value
−15 of y1 at x = −27.
Press 2nd ENTRY to display the last line again, use J to move the cursor to
the 2 and press 3 0 to have −30 → x: y1. Press ENTER for the approximate decimal
value −15.5361625298 of y1 at x = −30.
Press 2nd ENTRY to display the last line again, use J to move the cursor
to the minus sign and press 4 DEL DEL to have 4 → x: y1. Press ENTER for the
approximate decimal value 7.93700525984 of y1 at x = 4.
Press
the 4 and press
2nd ENTRY to display the last line again, use J to move the cursor to
1 2nd INS 0 to have 10 → x: y1. Press ENTER for the approximate
decimal value 10.7721734502 of y1 atx = 10.
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Getting started TI-85/86 calculators, 5
Repeat this process for the other two values in the table below. Only the values
at −27 and x = 8 are exact because only −27 and 8 of the
x-values are perfect cubes.
x y = 5x
.
1/3
= x y = 5x
1/3
.
=
−27 −15 −30 −15.5361625298
4 7.93700525984 10 10.7721734502
6 9.08560296416 8 10
Exercises
Use your calculator or computer to find the approximate decimal values of the expressions in Exercises T1
through T8. Do not simplify the expressions before entering them and be sure your machine is in radian
mode for the trigonometric function in Exercise T1. In some cases extra parentheses are needed to express
numerators, denominators, and expone nts.
O
1.
2.
3.
4.
5.
6.
7.
8.
(a)√6 cos(9/7) (b) 6 cos(π/73) (Edit the expression from part (a).)
O
(−5 − 1.63 × 10−2)
2 + 8
O
4 − 6
O
A + BCDwith A = 7, B = 6, C = 5, and D = 4 (Store the values first.)
O
1
log10(7)
2
1.34 × 106− 4 × 10
O
√
O
O
4 + 7
(−32)
5−1
− 3
7.12 × 10
8−10
4/5
−1
5
−8
Outlines of solutions
1a. 0.688885143177 • (If your result is 2.44887304686, your calculator is not in radian mode.) •
Press 2nd
ENTER for the answer.
√
6 cos ( 9 ÷ 7 ) to have the screen read
1b. 5.999748331 • If your last operation was the calculation of√6 cos (9/7), press 2nd ENTRY
to put it back on the screen. If you performed other calculations, type√6 cos (9/7) again. Press
√
6 cos (9/7). Then press
J until the c ursor is over the square root sign and press DEL to delete it. Move the cursor
to the 9 and press 2nd π to replace the 9 with π. Put the cursor on the close parenthesis and
press 2nd INS . Press ∧ 3 to insert ∧ 3 before the close parenthesis, so the screen reads
6 cos(π/7 ∧ 3), and press ENTER for the answer.
2. −0.199350118613 • Press ( (−) 5 − 1 . 6 3 EE (−) 2 ) 2nd x−1to display
(–5 − 1.63e–2)−1. (A e n stands for A × 10n.) Press ENTER for the answer. Notice that (−) is
for negation, − is for subtraction, and x−1is for taking reciprocals.
3. −86 • Use ( 2 + 8 ) ÷ ( 4 − 6 ) − 3 ∧ ( 5 − 1 ) to display
(2 + 8)/(4 − 6) − 3 ∧ (5 − 1). Press ENTER for the answer.
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Getting started TI-85/86 calculators, 6
4 3757 • To put t h e commands on one line for possible later editing, press 7 STOI A
ALPHA 2nd : 6 STOI B ALPHA 2nd : 5 STOI C ALPHA 2nd : 4 STOI
D ALPHA 2nd : ALPHA A + ALPHA B × ALPHA C ∧ ALPHA D so the screen
reads 7 → A : 6 → B : 5 → C : 4 → D : A+B ∗C ∧D. Press ENTER to store the values of A, B, C
and D and calculate the answer. The colon (above the period key) separates two commands on
one line. The ALPHA keys are needed b ecause STOI locks the calculator in upper-case alpha
mode.
5. 0.422549020007 • Use ( 1 ÷ 2 ) LOG ( 7 ) ENTER .
6. 1.3202247191 × 1013• Use ( 1 . 3 4 EE 6 − 4 EE 5 ) ÷ ( 7 . 1 2 EE
(–) 8 ) ENTER .
7. 2.00509554966 • Use 2nd
8. 16 • TI-85 and TI-86 calculators evaluate x
not always give the correct, real value of x
> 1. It is generally best t o write x
4 ÷ 5 ) ENTER give the wrong result(−16) since the negation is perfomed after the
√
( 4 + 7 ∧ ( 8 − 1 0 ) ) ENTER .
1/n
for negative x and odd integers n, but do
m/n
m/n
as (xm)
with negative x, odd n, and m an integer
1/n
for odd n. The keys
(-) 3 2 ∧ (
power. The symbols ( (-) 3 2 ) ∧ ( 4 ÷ 5 ) ENTER probably yield, on a TI85, (−12.94427191, 9.40456403668) representing −12.94427191 + 9.40456403668 i, which is the
approximate decimal value of one of the complex four-fifth roots of −32, rather than the real
four-fifth root 16. These symbols, however, give the correct result on a TI-86. The expressions
( (-) 3 2 ) ∧ 4 ∧ ( 1 ÷ 5 ) ENTER and ( (-) 3 2 ) ∧ ( 1 ÷ 5 )
∧ 4 ENTER give the correct result, since for any positive odd integer n and negative x, the
calculator interprets x ∧ (1/n) as the negative nth root of x.
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