270 Higher Engineering Mathematics
Hence, by cosine and sine rules,
iR=i 1 +i 2 = 26 .46sin(ωt+ 0. (^333) )A
Now try the following exercise
Exercise 109 Resultant phasors by the sine
and cosine rules
- Determine, using the cosine and sine rules, a
sinusoidal expression for:
y=2sinA+4cosA.
[4.5sin(A+ 63. 5 ◦)] - Givenv 1 =10sinωtvolts and
v 2 =14sin(ωt+π/ 3 )volts use the cosine and
sine rules to determine sinusoidal expressions
for (a)v 1 +v[ 2 (b)v 1 −v 2.
(a) 20.88sin(ωt+ 0. 62 )volts
(b) 12.50sin(ωt− 1. 33 )volts
]
In Problems 3 to 5, express the given expressions
in the formAsin(ωt±α)by using the cosine and
sine rules.
- 12sinωt+5cosωt
[13sin(ωt+ 0. 395 )] - 7sinωt+5sin
(
ωt+
π
4
)
[11.11sin(ωt+ 0. 324 )]
- 6sinωt+3sin
(
ωt−
π
6
)
[8.73sin(ωt− 0. 173 )]
25.5 Determining resultant phasors
by horizontal and vertical
components
If a right-angled triangle is constructed as shown in
Fig. 25.16, then 0ais called the horizontal component
ofFandabis called the vertical component ofF.
From trigonometry (see Chapter 11),
cosθ=
0 a
0 b
from which,
0 a= 0 bcosθ=Fcosθ
F
F sin
F cos a
b
0
Figure 25.16
i.e. the horizontal component ofF,H=Fcosθ
and sinθ=
ab
0 b
from whichab= 0 bsinθ
=Fsinθ
i.e. the vertical component ofF, V=Fsinθ
Determiningresultant phasorsbyhorizontal andvertical
components is demonstrated in the following worked
problems.
Problem 9. Two alternating voltages are given by
v 1 =15sinωtvolts andv 2 =25sin(ωt−π/6)
volts. Determine a sinusoidal expression for the
resultantvR=v 1 +v 2 by finding horizontal and
vertical components.
The relative positions ofv 1 andv 2 at timet=0are
shown in Fig. 25.17(a) and the phasor diagram is shown
in Fig. 25.17(b).
The horizontal component ofvR,
H=15cos0◦+25cos(− 30 ◦)= 0 a+ab= 36 .65V
The vertical component ofvR,
V=15sin0◦+25sin(− 30 ◦)=bc=− 12 .50V
Hence, vR=^0 c=
√
36. 652 +(− 12. 50 )^2
by Pythagoras’ theorem
=38.72 volts
tanφ=
V
H
=
− 12. 50
36. 65
=− 0. 3411
from which,φ=tan−^1 (− 0. 3411 )=− 18. 83 ◦
or − 0 .329 radians.
Hence, vR=v 1 +v 2 =^38 .72sin(ωt−^0.^329 )V
Problem 10. For the voltages in Problem 9,
determine the resultantvR=v 1 −v 2 using
horizontal and vertical components.