Understanding Engineering Mathematics

y

0 x

p/2

y = tan−^1 x

y

− 1 1 x

p

p/2

0

y = cos−^1 x

p

− 2

Figure 6.12Inverse cosine, cos−^1 xand inverse tan,tan−^1 x.

where PV is the principal value. Sometimes the principal value is distinguished by using
a capital initial letter, for example we might denote the PV of sin−^1 xby Sin−^1 x.

Solution to review question 6.1.5

In the range 0°≤θ≤ 90 °we have, referring to Figure 6.4:

(i) sin−^1

(

1

2

)

= 30 ° (ii) sin−^1

(√

3

2

)

= 60 °

(iii) cos−^1

(√

3

2

)

= 30 ° (iv) cos−^1

(

1

2

)

= 60 °

(v) tan−^1

(

1

√

3

)

= 30 ° (vi) tan−^1 (

√

3 )= 60 °

6.2.6 The Pythagorean identities – cos^2 Ysin^2 = 1

➤

172 195➤

The standard trigonometric identities are normally found on a formulae sheet. However,
the key identities should actually bememorised– and indeed should be second nature. In
fact, you only have to remember one or two, from which the rest may then be derived. A
sufficient minimal set to remember is in fact:

cos^2 θ+sin^2 θ= 1

and
sin(A+B)=sinAcosB+sinBcosA

If you have these at your fingertips most of the others are easily recalled or derived from
them. The first is the subject of this section, the second is treated in Section 6.2.7.
ThePythagorean identitiesare

cos^2 θ+sin^2 θ≡ 1

1 +tan^2 θ≡sec^2 θ

cot^2 θ+ 1 ≡cosec^2 θ

These can be obtained, as their name suggests, from Pythagoras’ theorem (154
➤
):