Higher Engineering Mathematics

178 GEOMETRY AND TRIGONOMETRY

5. Given cosA= 0 .42 and sinB= 0 .73 evaluate
   (a) sin(A−B), (b) cos(A−B), (c) tan (A+B),
   correct to 4 decimal places.
   [(a) 0.3136 (b) 0.9495 (c)−2.4687]
   In Problems 6 and 7, solve the equations for
   values ofθbetween 0◦and 360◦.


6. 3 sin(θ+ 30 ◦)=7 cosθ
   [64◦ 43 ′or 244◦ 43 ′]


7. 4 sin(θ− 40 ◦)=2 sinθ
   [67◦ 31 ′or 247◦ 31 ′]

18.2 Conversion ofasinωt+bcosωt

intoRsin(ωt+α)

(i)Rsin(ωt+α) represents a sine wave of maxi-

mum valueR, periodic time 2π/ω, frequency

ω/ 2 πand leadingRsinωtby angleα. (See

Chapter 15).

(ii)Rsin (ωt+α) may be expanded using the

compound-angle formula for sin(A+B), where

A=ωtandB=α. Hence,

Rsin (ωt+α)

=R[sinωtcosα+cosωtsinα]

=Rsinωtcosα+Rcosωtsinα

=(Rcosα) sinωt+(Rsinα) cosωt

(iii) Ifa=Rcosαandb=Rsinα, whereaandb

are constants, thenRsin(ωt+α)=asinωt+

bcosωt, i.e. a sine and cosine function of the

same frequency when added produce a sine

wave of the same frequency (which is further

demonstrated in Section 21.6).

(iv) Sincea=Rcosα, then cosα=a/R, and since

b=Rsinα, then sinα=b/R.

Figure 18.1

If the values ofaandbare known then the values ofR

andαmay be calculated. The relationship between

constantsa,b,Randαare shown in Fig. 18.1.

From Fig. 18.1, by Pythagoras’ theorem:

R=

√

a^2 +b^2

and from trigonometric ratios:

α=tan−^1 b/a

Problem 6. Find an expression for

3 sinωt+4 cosωt in the form Rsin(ωt+α)

and sketch graphs of 3 sinωt, 4 cosωt and

Rsin(ωt+α) on the same axes.

Let 3 sinωt+4 cosωt=Rsin (ωt+α)

then 3 sinωt+4 cosωt

=R[sinωtcosα+cosωtsinα]

=(Rcosα) sinωt+(Rsinα) cosωt

Equating coefficients of sinωtgives:

3 =Rcosα, from which , cosα=

3

R

Equating coefficients of cosωtgives:

4 =Rsinα, from which, sinα=

4

R

There is only one quadrant where both sinαand

cosαare positive, and this is the first, as shown in

Fig. 18.2. From Fig. 18.2, by Pythagoras’ theorem:

R=

√

(3^2 + 42 )= 5

Figure 18.2

From trigonometric ratios:α=tan−^143 = 53 ◦ 8 ′or

0.927 radians.