Understanding Engineering Mathematics

(やまだぃちぅ) #1
A 2 sin(ωt+α 2 ). When we come to complex numbers we will see how this sort of
thing can be done using objects calledphasors(371

), but really such methods
are simply shorthand for the ideas covered in this chapter. In particular we can
add two waves that are 90°out of phase using the results of Section 6.2.9, since
A 1 sin(ωt+ 90 °)+A 2 sin(ωt) is equivalent to acos(ωt)+bsin(ωt). For additions
such asA 1 sin(ωt+α)+A 2 sin(ωt),whereαis other than 90°, phasor methods are
equivalent to constructing a parallelogram with sidesA 1 andA 2 and included angleα
and taking the combined amplitude as the length of the diagonal,r, and the combined
phase to be the angle,θ, made by the diagonal with the sideA 2 (see Figure 6.15).

A 1 sin(ωt+α)+A 2 sin(ωt)=rsin(ωt+θ)

R
a q

A 1

A 2

Figure 6.15

Use the above methods to find the sine waves representing:

(i) 4 sin(ωt)+3cos(ωt) (ii) 6 sin(ωt)+4sin(ωt+ 45 °)

Answers to reinforcement exercises

6.3.1 Radian measure and the circle


A. (i)


π
5

(ii)

101 π
180

(iii)

2 π
3

(iv)

25 π
18

(v)

17 π
9

(vi) −

π
4

(vii) −

11 π
18

(viii) −

π
12

(ix)

3 π
20

(x)

91 π
60

B. (i) 120° (ii) 0° (iii) 270° (iv) 60°


(v) 30° (vi) 90° (vii) 40° (viii) 225°

(ix) 308° (x) 15°

C. (i)


π
3

,

2 π
3

(ii)

2 π
3

,

4 π
3

(iii) π,2π (iv)

4 π
3

,

8 π
3

(v) 2π,4π (vi)

8 π
3

,

16 π
3

(vii)

32 π
9

,

64 π
9

(viii) 4π,8π

Note: If you are adept with ratios you will have noticed that with the given radius the area
will always have a magnitude double that of the arc length.

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