Higher Engineering Mathematics, Sixth Edition

168 Higher Engineering Mathematics

i.e. θ+ 59. 03 ◦= 43. 32 ◦or 136. 68 ◦

Hence θ= 43. 32 ◦− 59. 03 ◦=− 15. 71 ◦

or θ= 136. 68 ◦− 59. 03 ◦= 77. 65 ◦

Since− 15. 71 ◦is the same as− 15. 71 ◦+ 360 ◦,i.e.

344. 29 ◦, then the solutions areθ= 77. 65 ◦or 344. 29 ◦,

which may be checked by substituting into the original

equation.

Problem 10. Solve the equation

3 .5cosA− 5 .8sinA= 6 .5for0◦≤A≤ 360 ◦.

Let 3.5cosA− 5 .8sinA=Rsin(A+α)

=R[sinAcosα+cosAsinα]

=(Rcosα)sinA+(Rsinα)cosA

Equating coefficients gives:

3. 5 =Rsinα,from which,sinα=

3. 5

R

and − 5. 8 =Rcosα,from which,cosα=

− 5. 8

R

There is only one quadrant in which both sine is posi-

tiveandcosine is negative, i.e. the second, as shown in

Fig. 17.7.

2708

908

(^18083608)
08


R
2 5.8
3.5
Figure 17.7
From Fig. 17.7,R=
√
[( 3. 5 )^2 +(− 5. 8 )^2 ]= 6 .774 and
θ=tan−^1
3. 5
5. 8
= 31. 12 ◦.
Henceα= 180 ◦− 31. 12 ◦= 148. 88 ◦.
Thus
3 .5cosA− 5 .8sinA= 6 .774sin(A+ 144. 88 ◦)= 6. 5
Hence sin(A+ 148. 88 ◦)=
6. 5
6. 774
,from which,
(A+ 148. 88 ◦)=sin−^1
6. 5
6. 774
= 73. 65 ◦or 106. 35 ◦
Thus A= 73. 65 ◦− 148. 88 ◦=− 75. 23 ◦
≡(− 75. 23 ◦+ 360 ◦)= 284. 77 ◦
or A= 106. 35 ◦− 148. 88 ◦=− 42. 53 ◦
≡(− 42. 53 ◦+ 360 ◦)= 317. 47 ◦
The solutions are thusA= 284. 77 ◦or 317. 47 ◦,which
may be checked in the original equation.
Now try the following exercise
Exercise 73 Further problems on the
conversion ofasinωt+bcosωtinto
Rsin(ω t+α)
In Problems 1 to 4, change the functions into the
formRsin(ωt±α).

1. 5sinωt+8cosωt [9.434sin(ωt+ 1. 012 )]


2. 4sinωt−3cosωt [5sin(ωt− 0. 644 )]


3. −7sinωt+4cosωt
   [8.062sin(ωt+ 2. 622 )]


4. −3sinωt−6cosωt
   [6.708sin(ωt− 2. 034 )]


5. Solve the followingequations for values ofθ
   between 0◦and 360◦:(a)2sinθ+4cosθ= 3
   (b) 12sinθ−9cosθ=7.
   [
   (a) 74. 44 ◦or 338. 70 ◦
   (b) 64. 69 ◦or 189. 05 ◦

]

6. Solve the following equations for
   0 ◦<A< 360 ◦:(a)3cosA+2sinA= 2. 8
   (b) 12 cosA−4sinA=11.
   [
   (a) 72. 73 ◦or 354. 63 ◦
   (b) 11. 15 ◦or 311. 98 ◦

]

7. Solve the followingequations for values ofθ
   between 0◦and360◦:(a)3sinθ+4cosθ= 3
   (b) 2cosθ+sinθ=2.
   [(a) 90◦or 343. 74 ◦(b) 0◦,53. 14 ◦]