Understanding Engineering Mathematics

question is given in terms of degrees/radians, then the answer should be given in the same
form)
f(θ+ 360 °)≡f(θ+ 2 π)=f(θ)

Thus, in particular

cos(θ+ 2 π)=cosθand tan(θ+ 2 π)=tanθ

This is reflected in the oscillatory nature of the graphs of the functions – the form of the
graph repeats itself at intervals of 2πfor sin and cos, and at intervals ofπfor tan. This
‘minimum repeating interval’ is called theperiodof the function. In general, if

f(x+p)=f(x)for allx

for some constantpthen we sayf(x)isperiodic with periodt. Of course, it also follows
that iff(x)has periodpthenf(x+np)=f(x)for any positive integern(providedf
is defined atx+np), but the term ‘period’ is usually reserved for the smallest value of
the period over which the function repeats. What, then, is theactualperiod of tanθ?
The cosine and sine functions are particularly important in engineering and science,
representing, as they do, general wave-like behaviour. Indeed if you think about the really
fundamental types of physical change that we commonly experience, they are either ‘expo-
nential’ growth or decay (127

➤
), or wave-like/oscillatory, which we can express in terms
of trig functions. Of course, in reality most oscillatory behaviour is ‘damped’, and this is
often expressed by multiplying the trig function by a decaying exponential.
The graphs of the trigonometric functions cosθ,sinθand tanθare shown below. cos
and sin repeat every cycle of 2πand oscillate between−1and+1. tan repeats every cycle
ofπ, and is ‘discontinuous’ (➤414) at odd multiples ofπ/2, tending to either−∞or
+∞. Also, notice that the cosine graph is simply the sine graph shifted along the axis by
π/2.

− 2 p −p p p p 3 p 2 p q

0

1

− 1

sin q

2 2 2

3 p

−− 2

Figure 6.8sinθ,− 2 π≤θ≤ 2 π.