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APPLIED MATHEMATICS FOR CIVIL ENGINEERING
H72N 35/ HR6L 48
CHAPTER 1
CURVES WITH POLAR EQUATIONS
In polar coordinates, the origin is called the pole, and the half-line for which the angle is zero
(equivalent to the positive x-axis) is called the polar axis. The coordinates of a point are
designated as (r,θ). The unit, radians will be used when measuring the value of θ.
To locate a point, determine terminal side θ first, then determine r.
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,
One difference between rectangular coordinates and polar coordinates is that, for each point in
the plane, there are limitless possibilities for the polar coordinates of that point.
,
,
,
,
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POLAR AND RECTANGULAR COORDINATES
Rectangular to Polar
Polar to rectangular
(
)
√
3
√
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Rectangular to Polar equation
A proton (positively charged) enters a magnetic field such that its path may be described by the
rectangular equation
, where measurements are in meters. Find the polar equation
of this circle.
Polar to Rectangular equation
Find the rectangular equation of the rose
.
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Exercises
1.
Find the polar equation of the circle
2.
Find the polar equation of each of the given rectangular equations.
3.
4.
.
[
(i)
[
(ii)
[
(iii)
[
(iv)
[
(v)
[
]
]
]
]
]
]
Find the rectangular equation of each of the given polar equations.
(i)
[
(ii)
[
(iii)
[
(iv)
[
(v)
[
]
]
]
]
]
Under certain conditions, the x- and y-components of a magnetic field B are given by the
equations
and
Write these equations in terms of polar coordinates.
[
5
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5.
The shape of a swimming pool can be described by the polar equation r = 5 cos θ
(dimensions in meters). Find the rectangular equation for the perimeter of the pool. State
the co-ordinates of the centre and radius of the circle.
(
6.
)
Part of a causeway is modelled by the polar equation
Convert it
to rectangular form and state the co-ordinates of the centre and radius of the locus
depicted. Find the arc length of this causeway. Given that
∫ √
7.
The control tower of an airport is taken to be at the pole, and the polar axis is taken as
due east in a polar coordinate graph. How far apart (in km) are planes, at the same
altitude, if their positions on the graph are 6.10, 1.25 and 8.45, 3.74?
[13.805 km]
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CURVES IN POLAR COORDINATES
Assume values of the independent variable θ and then find the corresponding values of the
dependent variable r. These points are then plotted and joined, thereby forming the curve that
represents the relation in polar coordinates.
Certain basic curves can be sketched directly from the equation.
Examples
1
The graph of the polar equation r = 3 is a circle of radius 3, with center at the pole.
2
The graph of
3
Plot the graph of
is a straight line through the pole.
. [ Cardioid ]
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4
Plot the graph of
5
Plot the graph of
. [ Limacon ]
. [ Rose ]
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6
Plot the graph of
. [Lemniscate ]
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Exercises
7.
Plot the curves of the given polar equations in polar coordinates.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
8.
An architect designs a patio shaped such that it can be described as the area within the
polar curve
, where measurements are in meters. Sketch the curve that
represents the perimeter of the patio.
9.
Plot the graph of
.
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Coordinate Systems
1
Rectangular coordinates
2
Cylindrical coordinates
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3
Spherical coordinates
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