Understanding Engineering Mathematics

r

x

y

q

We have

x^2 +y^2 =r^2

Dividing byr^2 gives

(x

r

) 2

+

(y

r

) 2

= 1

or cos^2 θ+sin^2 θ= 1

Thisidentity should definitely become second nature to you – it is absolutely vital. By
dividing through by cos^2 θwe get 1+tan^2 θ=sec^2 θand dividing through by sin^2 θgives
cot^2 θ+ 1 =cosec^2 θ, so there is no need to remember all three identities.

Solution to review question 6.1.6

(i) Using cos^2 θ+sin^2 θ=1wehave

cos^2 θ= 1 −sin^2 θ= 1 −

1

49

=

48

49

so

cosθ=

√

48

49

in first quadrant

=

4

√

3

7

Then tanθ=

sinθ

cosθ

=

1 / 7

4

√

3 / 7

=

1

4

√

3

(ii) sinθ=

√

1 −cos^2 θ=

√

1 −

1

3

=

√

2

3

tanθ=

√ 2 3 1 √ 3

=

√

2

(iii) sec^2 θ=

1

cos^2 θ

= 1 +tan^2 θ

= 1 +

(

1

√

2

) 2

=

3

2