Everything Maths Grade 12

9.4 CHAPTER 9. GEOMETRY

Polar co-ordinates

�

r

y

x

α

P

Notice that sinα =yr∴ y = r sinα

and cosα =xr∴ x = r cosα

so P can be expressed in two ways:

P (x;y) rectangular co-ordinates

or P (r;α) polar co-ordinates.

Compound angles

(See derivation of formulae in Chapter 12)

sin (α +β) = sinα cosβ + sinβ cosα

cos (α +β) = cosα cosβ− sinα sinβ

Now consider P�after a rotation of θ

P (x;y) = P (r cosα;r sinα)

P�(r cos (α +θ);r sin (α +θ))

Expand the co-ordinatesof P�

x− co-ordinate = r cos (α +θ)

= r [cosα cosθ− sinα sinθ]

= r cosα cosθ−r sinα sinθ

= x cosθ−y sinθ

y− co-ordinate = r sin (α +θ)

= r [sinα cosθ + sinθ cosα]

= r sinα cosθ +r cosα sinθ

= y cosθ +x sinθ

�

α

� P = (r cosα;r sinα)

P�

θ

which gives the formula P�= [(x cosθ−y sinθ;y cosθ +x sinθ)].

So to find the co-ordinates of P (1;

√

3) after a rotation of 45 ◦, we arrive at:

P� = [(x cosθ−y sinθ); (y cosθ +x sinθ)]

=

�

(1 cos 45◦−

√

3 sin 45◦); (

√

3 cos 45◦+ 1 sin 45◦)

�

=

��

1

√

2

−

√

3

√

2

�

;

�√

3

√

2

+

1

√

2

��

=

�

1 −

√

3

√

2

;

√

3 + 1

√

2

�

Exercise 9 - 9