Revision Test 1 : Basic arithmetic and fractions
4.3 Further calculator functions
4.3.1 Square and cube functions
Locate thex2andx3functions on your calculator and then check the following worked examples.
Problem 3. Evaluate 2.42 (i) Type in 2.4
(ii) Pressx2and 2.42appears on the screen.
(iii) Press=and the answer144
25 appears.
(iv) Press the S⇔D function and the fraction changes to a decimal 5.76
Alternatively, after step (ii) press Shift and =. Thus,2.42=5.76
Problem 4. Evaluate 0.172in engineering form (i) Type in 0.17
(ii) Pressx2and 0.172appears on the screen.
24 Basic Engineering Mathematics
(iii) Press Shift and=and the answer 0.0289 appears.
(iv) Press the ENG function and the answer changes to 28.9×10−3, which isengineering form.
Hence,0.172=28.9×10−3in engineering form. The ENG function is extremely important in engineering calculations.
Problem 5. Change 348620 into engineering form
(i) Type in 348620 (ii) Press=then ENG.
Hence,348620=348.62×103in engineering form.
Problem 6. Change 0.0000538 into engineering form
(i) Type in 0.0000538 (ii) Press=then ENG.
Hence,0.0000538=53.8×10−6in engineering form.
Problem 7. Evaluate 1.43 (i) Type in 1.4
(ii) Pressx3and 1.43appears on the screen.
(iii) Press=and the answer 343
125appears.
(iv) Press the S⇔D function and the fraction changes to a decimal: 2.744
Thus,1.43=2.744.
Now try the following Practice Exercise Practice Exercise 13 Square and cube functions (answers on page 341)
1. Evaluate 3.52 2. Evaluate 0.192
3. Evaluate 6.852correct to 3 decimal places.
4. Evaluate(0.036)2in engineering form.
5. Evaluate 1.5632 correct to 5 significant figures.
6. Evaluate 1.33
7. Evaluate 3.143 correct to 4 significant figures.
8. Evaluate(0.38)3correct to 4 decimal places.
9. Evaluate(6.03)3correct to 2 decimal places.
10. Evaluate(0.018)3in engineering form.
4.3.2 Reciprocal and power functions The reciprocal of 2 is1
2, the reciprocal of 9 is1 9 and the reciprocal ofxis1
x, which from indices may be written asx−1. Locate the reciprocal, i.e.x−1on the calculator.
Also, locate the power function, i.e. x, on your calculator and then check the following worked examples.
Problem 8. Evaluate 1 3.2 (i) Type in 3.2
(ii) Pressx−1and 3.2−1appears on the screen.
(iii) Press=and the answer 5
16 appears.
(iv) Press the S⇔D function and the fraction changes to a decimal: 0.3125
Thus, 1
3.2 =0.3125
Problem 9. Evaluate 1.55correct to 4 significant figures
(i) Type in 1.5
(ii) Pressxand 1.5appears on the screen.
(iii) Press 5 and 1.55appears on the screen.
(iv) Press Shift and = and the answer 7.59375 appears.
Thus,1.55=7.594 correct to 4 significant figures.
Problem 10. Evaluate 2.46−1.94correct to 3 decimal places
(i) Type in 2.4
(ii) Pressxand 2.4appears on the screen.
Using a calculator 25
(iii) Press 6 and 2.46appears on the screen.
(iv) The cursor now needs to be moved; this is achieved by using the cursor key (the large blue circular function in the top centre of the calculator). Press→
(v) Press−
(vi) Type in 1.9, pressx, then press 4.
(vii) Press=and the answer 178.07087...appears.
Thus, 2.46−1.94=178.071 correct to 3 decimal places.
Now try the following Practice Exercise
Practice Exercise 14 Reciprocal and power functions (answers on page 341)
1. Evaluate 1
1.75 correct to 3 decimal places.
2. Evaluate 1 0.0250 3. Evaluate 1
7.43correct to 5 significant figures.
4. Evaluate 1
0.00725correct to 1 decimal place.
5. Evaluate 1
0.065− 1
2.341correct to 4 signifi-cant figures.
6. Evaluate 2.14
7. Evaluate (0.22)5 correct to 5 significant figures in engineering form.
8. Evaluate (1.012)7 correct to 4 decimal places.
9. Evaluate(0.05)6in engineering form.
10. Evaluate 1.13+2.94−4.42correct to 4 sig-nificant figures.
4.3.3 Root and×10xfunctions Locate the square root function √
and the √
function (which is a Shift function located above thex function) on your calculator. Also, locate the
×10x function and then check the following worked examples.
Problem 11. Evaluate√ 361 (i) Press the√
function.
(ii) Type in 361 and√
361 appears on the screen.
(iii) Press=and the answer 19 appears.
Thus,√
361=19.
Problem 12. Evaluate√4 81 (i) Press the √
function.
(ii) Type in 4 and√4
appears on the screen.
(iii) Press→to move the cursor and then type in 81 and√4
81 appears on the screen.
(iv) Press=and the answer 3 appears.
Thus,√4 81=3.
Problem 13. Evaluate 6×105×2×10−7 (i) Type in 6
(ii) Press the×10xfunction (note, you do not have to use×).
(iii) Type in 5 (iv) Press×
(v) Type in 2
(vi) Press the×10x function.
(vii) Type in−7
(viii) Press=and the answer 3
25appears.
(ix) Press the S⇔D function and the fraction changes to a decimal: 0.12
Thus,6×105×2×10−7=0.12
Now try the following Practice Exercise Practice Exercise 15 Root and×10x functions (answers on page 341)
1. Evaluate√
4.76 correct to 3 decimal places.
2. Evaluate √
123.7 correct to 5 significant figures.
26 Basic Engineering Mathematics
3. Evaluate√
34528 correct to 2 decimal places.
4. Evaluate √
0.69 correct to 4 significant figures.
5. Evaluate√
0.025 correct to 4 decimal places.
6. Evaluate√3
17 correct to 3 decimal places.
7. Evaluate √4
773 correct to 4 significant figures.
8. Evaluate√5
3.12 correct to 4 decimal places.
9. Evaluate √3
0.028 correct to 5 significant figures.
10. Evaluate√6
2451−√4
46 correct to 3 decimal places.
Express the answers to questions 11 to 15 in engineering form.
11. Evaluate 5×10−3×7×108 12. Evaluate 3×10−4
8×10−9
13. Evaluate 6×103×14×10−4 2×106
14. Evaluate 56.43×10−3×3×104
8.349×103 correct to 3 decimal places.
15. Evaluate99×105×6.7×10−3
36.2×10−4 correct to 4 significant figures.
4.3.4 Fractions Locate the
and
functions on your calculator (the latter function is a Shift function found above the
function) and then check the following worked examples.
Problem 14. Evaluate1 4+2
3 (i) Press the
function.
(ii) Type in 1
(iii) Press↓on the cursor key and type in 4
(iv) 1
4 appears on the screen.
(v) Press→on the cursor key and type in+ (vi) Press the
function.
(vii) Type in 2
(viii) Press↓on the cursor key and type in 3 (ix) Press→on the cursor key.
(x) Press=and the answer 11
12 appears.
(xi) Press the S⇔D function and the fraction changes to a decimal 0.9166666...
Thus, 1 4+2
3 =11
12=0.9167 as a decimal, correct to 4 decimal places.
It is also possible to deal withmixed numberson the calculator. Press Shift then the
function and
appears.
Problem 15. Evaluate 51 5−33
4 (i) Press Shift then the
function and
appears on the screen.
(ii) Type in 5 then→on the cursor key.
(iii) Type in 1 and↓on the cursor key.
(iv) Type in 5 and 51
5 appears on the screen.
(v) Press→on the cursor key.
(vi) Type in – and then press Shift then the
function and 51
5−
appears on the screen.
(vii) Type in 3 then→on the cursor key.
(viii) Type in 3 and↓on the cursor key.
(ix) Type in 4 and 51 5−33
4 appears on the screen.
(x) Press=and the answer 29
20 appears.
(xi) PressS⇔Dfunction and the fraction changes to a decimal 1.45
Thus,51 5−33
4=29 20=1 9
20 =1.45as a decimal.
Using a calculator 27
Now try the following Practice Exercise
Practice Exercise 16 Fractions (answers on page 341)
1. Evaluate 4 5−1
3 as a decimal, correct to 4 decimal places.
2. Evaluate2 3−1
6+3
7 as a fraction.
3. Evaluate 25 6+15
8 as a decimal, correct to 4 significant figures.
4. Evaluate 56 7−31
8 as a decimal, correct to 4 significant figures.
5. Evaluate1 3−3
4× 8
21 as a fraction.
6. Evaluate3 8+5
6−1
2 as a decimal, correct to 4 decimal places.
7. Evaluate3 4×4
5−2 3÷4
9 as a fraction.
8. Evaluate 88 9÷22
3 as a mixed number.
9. Evaluate 31 5×11
3−1 7
10 as a decimal, cor-rect to 3 decimal places.
10. Evaluate
41 5−12
3
31 4×23
5 −2
9as a decimal, cor-rect to 3 significant figures.
4.3.5 Trigonometric functions
Trigonometric ratios will be covered in Chapter 21.
However, very briefly, there are three functions on your calculator that are involved with trigonometry. They are:
sinwhich is an abbreviation ofsine coswhich is an abbreviation ofcosine, and tanwhich is an abbreviation oftangent
Exactly what these mean will be explained in Chapter 21.
There are two main ways that angles are measured, i.e.
indegrees or inradians. Pressing Shift, Setup and 3 shows degrees, and Shift, Setup and 4 shows radians.
Press 3 and your calculator will be indegrees mode, indicated by a small D appearing at the top of the screen.
Press 4 and your calculator will be inradian mode, indicated by a small R appearing at the top of the screen.
Locate the sin, cos and tan functions on your calculator and then check the following worked examples.
Problem 16. Evaluate sin 38◦
(i) Make sure your calculator is in degrees mode.
(ii) Press sin function and sin( appears on the screen.
(iii) Type in 38 and close the bracket with) and sin (38) appears on the screen.
(iv) Press=and the answer 0.615661475...appears.
Thus,sin 38◦=0.6157,correct to 4 decimal places.
Problem 17. Evaluate 5.3 tan (2.23 rad)
(i) Make sure your calculator is in radian mode by pressing Shift then Setup then 4 (a small R appears at the top of the screen).
(ii) Type in 5.3 then press tan function and 5.3 tan(
appears on the screen.
(iii) Type in 2.23 and close the bracket with) and 5.3 tan (2.23) appears on the screen.
(iv) Press=and the answer−6.84021262...appears.
Thus,5.3 tan(2.23 rad)= −6.8402,correct to 4 dec-imal places.
Now try the following Practice Exercise Practice Exercise 17 Trigonometric functions (answers on page 341)
Evaluate the following, each correct to 4 decimal places.
1. Evaluate sin 67◦ 2. Evaluate cos 43◦ 3. Evaluate tan 71◦ 4. Evaluate sin 15.78◦ 5. Evaluate cos 63.74◦
6. Evaluate tan 39.55◦−sin 52.53◦ 7. Evaluate sin(0.437 rad)
28 Basic Engineering Mathematics
8. Evaluate cos(1.42 rad) 9. Evaluate tan(5.673 rad) 10. Evaluate (sin 42.6◦)(tan 83.2◦)
cos 13.8◦
4.3.6 πandexfunctions
Press Shift and then press the×10xfunction key andπ appears on the screen. Either press Shift and=(or= andS⇔D) and the value ofπappears in decimal form as 3.14159265...
Press Shift and then press the ln function key ande appears on the screen. Enter 1 and then press=and e1=e=2.71828182...
Now check the following worked examples involvingπ andex functions.
Problem 18. Evaluate 3.57π (i) Enter 3.57
(ii) Press Shift and the×10xkey and 3.57πappears on the screen.
(iii) Either press Shift and = (or = and S⇔D) and the value of 3.57π appears in decimal as 11.2154857...
Hence,3.57π=11.22 correct to 4 significant figures.
Problem 19. Evaluatee2.37
(i) Press Shift and then press the ln function key and eappears on the screen.
(ii) Enter 2.37 ande2.37appears on the screen.
(iii) Press Shift and=(or=andS⇔D) and the value ofe2.37appears in decimal as 10.6973922...
Hence,e2.37=10.70 correct to 4 significant figures.
Now try the following Practice Exercise Practice Exercise 18 πandexfunctions (answers on page 341)
Evaluate the following, each correct to 4 significant figures.
1. 1.59π 2. 2.7(π−1)
3. π2 √ 13−1
4. 3eπ 5. 8.5e−2.5 6. 3e2.9−1.6 7. 3e(2π−1) 8. 2πeπ3
9.
5.52π 2e−2×√
26.73
10.
⎡
⎣ e 2−
√3
π×√ 8.57
⎤
⎦