Methods of adding alternating waveforms 269

y 155

/6 or 30 8

y 254

Figure 25.12
y 155

y 25

4

(^0) 
yR
a
b
308
Figure 25.13
Usingthecosineruleontriangle0abofFig.25.13gives:
y^2 R= 52 + 42 −[2( 5 )( 4 )cos 150◦]
= 25 + 16 −(− 34. 641 )
= 75. 641
from which, yR=
√
75. 641 = 8. 697
Using the sine rule,
8. 697
sin150◦

4
sinφ
from which, sinφ=
4sin150◦
8. 697
= 0. 22996
and φ=sin−^10. 22996
= 13. 29 ◦or 0.232 rad
Hence, yR=y 1 +y 2 =5sinωt+4sin(ωt−π/ 6 )
= 8 .697sin(ωt− 0. 232 )
Problem 7. Giveny 1 =2sinωtand
y 2 =3sin(ωt+π/ 4 ), obtain an expression, by
calculation, for the resultant,yR=y 1 +y 2.
When timet=0, the position of phasorsy 1 and y 2
are as shown in Fig. 25.14(a). To obtain the resul-
tant,y 1 is drawn horizontally, 2 units long,y 2 is drawn
3 units long at an angle ofπ/4 rads or 45◦and joined to
the end ofy 1 as shown in Fig. 25.14(b).
From Fig. 25.14(b), and using the cosine rule:
yR^2 = 22 + 32 −[2( 2 )( 3 )cos135◦]
= 4 + 9 −[− 8 .485]= 21. 49
Hence, yR=
√
21. 49 = 4. 6357
y 152 y 152
y 253 y 253
yR
/4 or 45 8 ^1358458
(a) (b)
Figure 25.14
Using the sine rule:^3
sinφ

4. 6357
   sin135◦
   from which, sinφ=3sin135
   ◦


5. 
   

   6357
   = 0. 45761
   Hence, φ=sin−^10. 45761
   = 27. 23 ◦or 0.475 rad.
   Thus, by calculation, yR=^4 .635sin(ωt+^0.^475 )
   Problem 8. Determine
   20sinωt+10sin
   (
   ωt+
   π
   3
   )
   using the cosine
   and sine rules.
   From the phasor diagram of Fig. 25.15, and using the
   cosine rule:
   i^2 R= 202 + 102 −[2( 20 )( 10 )cos 120◦]
   = 700
   Hence,iR=
   √
   700 = 26 .46 A
   i 25 10 A
   i 15 20 A
   iR
   ^608
   Figure 25.15
   Using the sine rule gives :
   10
   sinφ

   
   


6. 46
   sin120◦
   from which, sinφ=
   10sin120◦


7. 46
   = 0. 327296
   and φ=sin−^10. 327296 = 19. 10 ◦
   = 19. 10 ×
   π
   180
   = 0 .333 rad