2.3 Redes inal´ambricas de sensores
3.1.4 Algoritmos de cifrado
One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. (For more on circles, see Chapter 26.)
With reference to Figure 20.2, for arc length s, θradians=s
r
r
r S
O
Figure 20.2
When s is the whole circumference, i.e. when s=2πr,
θ=s r =2πr
r =2π
In one revolution,θ=360◦. Hence, the relationship betweendegrees and radiansis
360◦=2πradians or180◦=πrad i.e. 1 rad=180◦
π ≈57.30◦
Here are some worked examples on angular measure-ment.
Problem 1. Evaluate 43◦29+27◦43 43◦29
+ 27◦43 71◦12
1◦
(i) 29+43=72
(ii) Since 60=1◦,72=1◦12
(iii) The 12is placed in the minutes column and 1◦is carried in the degrees column.
(iv) 43◦+27◦+1◦(carried)=71◦. Place 71◦in the degrees column.
This answer can be obtained using thecalculator as follows.
1. Enter 43 2. Press◦’ ’ ’ 3. Enter 29 4. Press◦’ ’ ’ 5. Press+ 6. Enter 27 7. Press◦’ ’ ’ 8. Enter 43 9. Press◦’ ’ ’ 10. Press= Answer=71◦12
Thus,43◦29+27◦43=71◦12. Problem 2. Evaluate 84◦13−56◦39
84◦13
− 56◦39 27◦34 (i) 13−39cannot be done.
(ii) 1◦or 60is ‘borrowed’ from the degrees column, which leaves 83◦in that column.
(iii) (60+13)−39=34, which is placed in the minutes column.
(iv) 83◦−56◦=27◦, which is placed in the degrees column.
This answer can be obtained using thecalculator as follows.
1. Enter 84 2. Press◦’ ’ ’ 3. Enter 13 4. Press◦’ ’ ’ 5. Press− 6. Enter 56 7. Press◦’ ’ ’ 8. Enter 39 9. Press◦’ ’ ’ 10. Press= Answer=27◦34
Thus,84◦13−56◦39=27◦34.
Problem 3. Evaluate 19◦5147+63◦2734 19◦5147
+ 63◦2734 83◦1921
1◦ 1
Angles and triangles 167
(i) 47+34=81
(ii) Since 60=1,81=121
(iii) The 21is placed in the seconds column and 1 is carried in the minutes column.
(iv) 51+27+1=79 (v) Since 60=1◦,79=1◦19
(vi) The 19is placed in the minutes column and 1◦ is carried in the degrees column.
(vii) 19◦+63◦+1◦(carried)=83◦. Place 83◦in the degrees column.
This answer can be obtained using the calculatoras follows.
1. Enter 19 2. Press◦’ ’ ’ 3. Enter 51 4. Press◦’ ’ ’ 5. Enter 47 6. Press◦’ ’ ’ 7. Press+ 8. Enter 63 9. Press◦’ ’ ’ 10. Enter 27 11. Press◦’ ’ ’
12. Enter 34 13. Press◦’ ’ ’ 14. Press= Answer=83◦1921
Thus,19◦5147+63◦2734=83◦1921. Problem 4. Convert 39◦27to degrees in decimal form
39◦27=3927◦ 60 27◦
60 =0.45◦ by calculator Hence, 39◦27=3927◦
60 =39.45◦
This answer can be obtained using the calculatoras follows.
1. Enter 39 2. Press◦’ ’ ’ 3. Enter 27 4. Press◦’ ’ ’ 5. Press= 6. Press◦’ ’ ’ Answer=39.45◦
Problem 5. Convert 63◦2651to degrees in decimal form, correct to 3 decimal places
63◦2651=63◦2651
60 =63◦26.85 63◦26.85=6326.85◦
60 =63.4475◦ Hence, 63◦2651=63.448◦ correct to 3 decimal places.
This answer can be obtained using the calculatoras follows.
1. Enter 63 2. Press◦’ ’ ’ 3. Enter 26 4. Press◦’ ’ ’ 5. Enter 51 6. Press◦’ ’ ’ 7. Press = 8. Press◦’ ’ ’ Answer=63.4475◦
Problem 6. Convert 53.753◦to degrees, minutes and seconds
0.753◦=0.753×60=45.18 0.18=0.18×60=11
to the nearest second Hence, 53.753◦=53◦4511
This answer can be obtained using the calculator as follows.
1. Enter 53.753 2. Press=
3. Press◦’ ’ ’ Answer=53◦4510.8 Now try the following Practice Exercise
Practice Exercise 76 Angular measurement (answers on page 348)
1. Evaluate 52◦39+29◦48 2. Evaluate 76◦31−48◦37
3. Evaluate 77◦22+41◦36−67◦47 4. Evaluate 41◦3716+58◦2936 5. Evaluate 54◦3742−38◦5325
6. Evaluate 79◦2619−45◦5856+53◦2138 7. Convert 72◦33to degrees in decimal form.
8. Convert 27◦4515 to degrees correct to 3 decimal places.
9. Convert 37.952◦to degrees and minutes.
10. Convert 58.381◦ to degrees, minutes and seconds.
Here are some further worked examples on angular measurement.
Problem 7. State the general name given to the following angles: (a) 157◦(b) 49◦(c) 90◦(d) 245◦ (a) Any angle between 90◦ and 180◦ is called an
obtuse angle.
Thus,157◦is an obtuse angle.
(b) Any angle between 0◦and 90◦is called an acute angle.
168 Basic Engineering Mathematics
Thus,49◦is an acute angle.
(c) An angle equal to 90◦is called aright angle.
(d) Any angle greater than 180◦and less than 360◦is called a reflex angle.
Thus,245◦is a reflex angle.
Problem 8. Find the angle complementary to 48◦39
If two angles add up to 90◦ they are called comple-mentary angles. Hence,the angle complementary to 48◦39is
90◦−48◦39=41◦21
Problem 9. Find the angle supplementary to 74◦25
If two angles add up to 180◦ they are called supple-mentary angles. Hence, the angle supplementary to 74◦25is
180◦−74◦25=105◦35 Problem 10. Evaluate angleθin each of the diagrams shown in Figure 20.3
(a) (b)
418
538 448
Figure 20.3
(a) The symbol shown in Figure 20.4 is called aright angleand it equals 90◦.
Figure 20.4
Hence, from Figure 20.3(a), θ+41◦=90◦
from which, θ =90◦−41◦=49◦
(b) An angle of 180◦lies on a straight line.
Hence, from Figure 20.3(b), 180◦=53◦+θ+44◦
from which, θ =180◦−53◦−44◦=83◦
Problem 11. Evaluate angleθin the diagram shown in Figure 20.5
39⬚
58⬚ 64⬚ 108⬚
Figure 20.5
There are 360◦in a complete revolution of a circle.
Thus, 360◦=58◦+108◦+64◦+39◦+θ
from which, θ=360◦−58◦−108◦−64◦−39◦=91◦ Problem 12. Two straight linesABandCD intersect at 0. If∠AOCis 43◦, find∠AOD,
∠DOBand∠BOC
From Figure 20.6,∠AODis supplementary to∠AOC.
Hence,∠AOD=180◦−43◦=137◦.
When two straight lines intersect, the vertically opposite angles are equal.
Hence, ∠DOB=43◦and∠BOC137◦
A D
C B
438 0
Figure 20.6
Angles and triangles 169
Problem 13. Determine angleβin Figure 20.7
1338
␣

Figure 20.7
α=180◦−133◦=47◦(i.e. supplementary angles).
α=β=47◦ (corresponding angles between parallel lines).
Problem 14. Determine the value of angleθin Figure 20.8
23⬚37⬘
35⬚49⬘ A
F
C
B E G
D
Figure 20.8
Let a straight lineFGbe drawn throughEsuch thatFG is parallel toABandCD.
∠BAE=∠AEF(alternate angles between parallel lines ABandFG), hence∠AEF=23◦37.
∠ECD=∠FEC(alternate angles between parallel lines FGandCD), hence∠FEC=35◦49.
Angleθ=∠AEF+∠FEC=23◦37+35◦49
=59◦26
Problem 15. Determine anglescanddin Figure 20.9
d
b a
c
468
Figure 20.9
a=b=46◦ (corresponding angles between parallel lines).
Also,b+c+90◦=180◦(angles on a straight line).
Hence, 46◦+c+90◦=180◦, from which,c=44◦. banddare supplementary, henced=180◦−46◦
=134◦.
Alternatively, 90◦+c=d(vertically opposite angles).
Problem 16. Convert the following angles to radians, correct to 3 decimal places.
(a) 73◦(b) 25◦37
Although we may be more familiar with degrees, radians is the SI unit of angular measurement in engineering (1 radian≈57.3◦).
(a) Since 180◦=πrad then 1◦= π 180rad.
Hence,73◦=73× π
180rad=1.274 rad.
(b) 25◦37=2537◦
60 =25.616666...
Hence, 25◦37=25.616666...◦
=25.616666...× π 180rad
=0.447 rad.
Problem 17. Convert 0.743 rad to degrees and minutes
Since 180◦=πrad then 1 rad=180◦ π Hence,0.743 rad=0.743×180◦
π =42.57076...◦
=42◦34 Since π rad=180◦, then π
2rad=90◦, π
4rad=45◦, π
3rad=60◦andπ
6rad=30◦
Now try the following Practice Exercise Practice Exercise 77 Further angular measurement (answers on page 348)
1. State the general name given to an angle of 197◦.
2. State the general name given to an angle of 136◦.
3. State the general name given to an angle of 49◦.
170 Basic Engineering Mathematics
4. State the general name given to an angle of 90◦.
5. Determine the angles complementary to the following.
(a) 69◦ (b) 27◦37 (c) 41◦343 6. Determine the angles supplementary to
(a) 78◦ (b) 15◦ (c) 169◦4111
7. Find the values of angle θ in diagrams (a) to (i) of Figure 20.10.
1208 (a)
558 508 (c)
608 808 (f)
(e) 708
1508
378
(d)
(g) 50.78
(h) 1118
688 458 578
(i) 368
708 ( b)
Figure 20.10
8. With reference to Figure 20.11, what is the name given to the lineXY? Give examples of each of the following.
x
y
1 4
5 8 3 2
7 6
Figure 20.11
(a) vertically opposite angles (b) supplementary angles (c) corresponding angles (d) alternate angles
9. In Figure 20.12, find angleα.
137⬚29⬘
16⬚49⬘
␣
Figure 20.12
10. In Figure 20.13, find anglesa,bandc.
c a b 298
698
Figure 20.13
11. Find angleβin Figure 20.14.
133⬚
98⬚

Figure 20.14
12. Convert 76◦to radians, correct to 3 decimal places.
13. Convert 34◦40to radians, correct to 3 deci-mal places.
14. Convert 0.714 rad to degrees and minutes.