The Chemistry Maths Book, Second Edition

72 Chapter 3Transcendental functions

This is one of the simplest second-order differential equations (Chapter 12) and a

solution is

x(t) 1 = 1 A 1 cos 1 ωt

where A, the amplitude, is the maximum displacement from equilibrium and

is called the angular frequency. The displacementx(t)is periodic with

respect to time, with period

where is the frequency of vibration. A plot of displacement against time

is very similar to that in Figure 3.12 (see Figure 12.3).

3.3 Inverse trigonometric functions

Ify 1 = 1 sin 1 xthen xis the angle whose sine is y, and is given by the inverse sine function

sin

− 1

y:

ify 1 = 1 sin 1 x thenx 1 = 1 sin

− 1

y (3.15)

Because of the possible confusion between the notation for the inverse sine,

sin

− 1

y, and the inverse of the sine, (sin 1 y)

− 1

, an alternative notation for the inverse

trigonometric functions is often used:

arcsin 1 y 1 = 1 sin

− 1

y, arccos 1 y 1 = 1 cos

− 1

y, arctan 1 y 1 = 1 tan

− 1

y (3.16)

The inverse functions are multi-valued functions; for example, as indicated in

Figure 3.13, many angles have the same sine:

sin x 1 = 1 sin 1 (π 1 − 1 x) 1 = 1 sin 1 (x 1 ± 12 nπ) (3.17)

ν=

1

2 π

k

m

τ

ων

==

21 π

ω= km

+1

− 1

o

x

y

•• • y=sinx

xπ−x 2 π+x

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Figure 3.13