76 Chapter 3Transcendental functions
EXAMPLE 3.14Express and in terms ofsin 1 θandcos 1 θ.
From equations (3.21),
Figure 3.11 shows that and (see also Example 3.5). Therefore
Similarly, using equations (3.22),
0 Exercises 24
The expressions for the sum and difference of angles are important for the calculation
of integrals (see Chapter 6) and for the description of the combination (interference)
of waves.
EXAMPLE 3.15The harmonic wave travelling in the x-direction described in
Example 3.7 and shown in Figure 3.12 has wave function
The same wave travelling in the opposite direction is (replacing tby −t)
As the waves overlap they interfere to give a new wave whose wave function is a linear
combination of the form
ψ 1 = 1 aφ
+1 + 1 bφ
−=−
++
aA
x
tbA
x
sin 22 ππsin t
λ
ν
λ
ν
φ
λ
ν
−=+
A
x
sin2π t
φ
λ
ν
+=−
A
x
sin2π t
cos cos cos sin sin sin
πππ
222
±
θθθθ==∓∓
sin cos
π
2
±
θθ=
cos
π
2
sin = 0
π
2
= 1
sin sin cos cos sin
ππ π
22 2
±
θθθ=±
cos
π
2
±
θ
sinπ
2
±
θ