Understanding Engineering Mathematics

6.2.5 Inverse trigonometric functions

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172 195➤

The inverse function (100

➤
)ofsinxis denoted sin−^1 x(Sometimes the notation ‘arcsin
x’ is used). That is ify=sinxthenx=sin−^1 y.Sosin−^1 xisthe angle whose sine is
x. Similarly

cos−^1 x=angle whose cosine isx

tan−^1 x=angle whose tangent isx

sec−^1 x=angle whose sec isx

cosec−^1 x=angle whose cosec isx

cot−^1 x=angle whose cot isx

Note that because, for example,

sin 135°=sin 45°=

1

√

2

the quantity sin−^1

(

1

√

2

)

can take more than one value – we say it ismulti-valued.In

general, because of periodicity all the inverse trigonometric functions are multi-valued.
Thus, for a givenx,sin−^1 xwill yield an infinite range of values. In order to restrict to a
unique value of sin−^1 xfor each value ofxwe take what is called theprincipal valueof
sin−^1 x, which is the value lying in the range

−

π

2

≤sin−^1 x≤

π

2

So, for example, the principal value of sin−^1

(

1

√

2

)

is 45°. This is shown in Figure 6.11.

y

− 1 0 1 x

p/2 y = sin− (^1) x
p
− 2
Figure 6.11Inverse sine, sin−^1 x.
Similarly we can define principal values for the other inverse trigonometric func-
tions – Figure 6.12 illustrates the principal value ranges for cos−^1 xand tan−^1 x.
Since the inverse functions are multi-valued we need expressions for the general solution
of equations such as siny=x, i.e. we need to obtain expressions for the most general
angle given by sin−^1 xand other inverse functions. We will simply state the results here:
sin−^1 x=nπ+(− 1 )nPV
cos−^1 x= 2 nπ±PV
tan−^1 x=nπ+PV