The Chemistry Maths Book, Second Edition

3.5 Polar coordinates 79

converting from cartesian coordinates to polar coordinates. The correct value of θis

determined by the quadrant in which the point(x, y)lies. Whenx 1 > 10 , the point lies

in the first or fourth quadrant (see Figure 3.6), and the angle is the principal value,

θ 1 = 1 tan

− 1

(y 2 x). Whenx 1 < 10 , the point lies in the second or third quadrant, and the

angle isθ 1 = 1 [principal value 1 + 1 π]. Therefore,

(3.28)

EXAMPLE 3.17Find the polar coordinates (r, 1 θ)of the point whose cartesian

coordinates are(x, y) 1 = 1 (3, 4).

We have

r

2

1 = 1 x

2

1 + 1 y

2

1 = 1 25, r 1 = 15

tan 1 θ 1 = 1 y 2 x 1 = 142 3, θ 1 = 1 tan

− 1

(4 2 3) 1 ≈ 153 °

The point (3, 4) lies in the first quadrant, and the angle is the principal value of the

inverse tangent.

0 Exercises 29

EXAMPLE 3.18Find the polar coordinates (r, θ)of the point whose cartesian

coordinates are(x, y) 1 = 1 (−1, 2).

We have

Use of the inverse tangent facility of a pocket calculator gives the principal value,

tan

− 1

(−2) 1 ≈ 1 − 63 °. But the point(−1, 2)lies in the second quadrant, where 90 ° 1 < 1 θ 1

< 1180 °, and the correct angle is

tan

− 1

(−2) 1 + 1 π 1 ≈ 1 − 63 ° 1 + 1180 ° 1 = 1117 °

with the same tangent value.

0 Exercises 30

0 Exercise 31

rxy r

y

x

222

1

55

22

=+= =+

==− = −+

−

,

tanθθ , tan ( ) π

θ=













+<

−

tan

1

0

y

x

π when x

rxy

y

x

=+ + = x











 >

22 − 1

,tanθ when 0