082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

4.4 Polar coordinates 161
y
O
u
r
x
Figure4.11
Thepolar coordinates arer, θ;the
Cartesian coordinates arex,y.
P
y
A
x
–3.4641
4
y
210°
–2
Figure4.12
The Cartesian coordinates canbe
calculated from the polar coordinates.
x
Example4.5 A point has polar coordinatesr = 4, θ = 210 ◦ . Calculate the Cartesian coordinates of
the point.Plot the point.
Solution TheCartesian coordinatesare given by
x =rcos θ = 4cos210 ◦ = −3.4641
y=rsinθ =4sin210 ◦ =−2
Figure 4.12 illustratesthe point.
We now look atthe problemofcalculatingthe polarcoordinates given the Cartesian
coordinates.Equations(4.1)and(4.2)canbearrangedsothatrand θ canbefoundfrom
the values ofxandy. Consider a typical pointPasshown inFigure 4.11.
The Cartesian coordinates are (x,y). Suppose that the values ofxandyare known.
The polar coordinates are r,θ; these values are unknown. By applying Pythagoras’s
theoremto △OPA wesee that
and so
r 2 =x 2 +y 2
r = √ x 2 +y 2
Note that since r is the distance from O to P it is always positive and so the positive
square rootistaken.
We now express θ in terms of the Cartesian coordinatesxandy. From Figure 4.11
wesee that
tanθ = y x
and hence
)
θ = tan −1 ( y
x
Insummary wehave
r = √ x 2 +y 2
θ = tan −1 ( y
x
)
( ) y
However, we need to exercise a little extra care before calculating tan −1 and
x
reading the resultfrom a calculator. As an illustrationnote that
tan40 ◦ = 0.8391 and tan220 ◦ = 0.8391