Introduction to Polar Coordinates: A Different Coordinate System
Exploring the concept of polar coordinates, a coordinate system introduced in the mid-seventeenth century by notable mathematicians independently. Learn about measuring angles in radians, converting between degrees and radians, and drawing graphs in polar coordinates by considering length and angle values.
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Polar Coordinates Stephen Earles ste9@aber.ac.uk s.n.f.earles@swansea.ac.uk
So far when plotting points on a graph we have used cartesian coordinates (named after ??? ? ????????? (1596 1650) the French philosopher and mathematician) in the ? ??? ? direction.
? ? ? ?
From the FM 4 course (which is an A2 course) you will look at a different coordinate system known as ????? ???????????
These concepts were introduced in the mid-seventeenth century by ?? ?????? ?? ????? ??????? ??? ??????????? ????????? independently. With this system of coordinates, you work with a distance from a reference point (usually the origin) called the ????. You also need to measure the angle from a reference direction (usually the positive x-axis) called the ??????? ????.
These angles can be measured in terms of ??????? or in terms of ??????? . We can convert between degrees and radians as follows, 360 = 2? ???? 180 = ? ???? 90 = 30 = ?2 ?6
Convert the following angles from degrees into radians. ?) 45 ?) 60 ?) 120 ?) 135 ?) 150 ?) 210 ?) 225 ) 240 ?) 270 ?) 300 ?) 315 ?) 330
Radians are normally used to measure angles in the A level course. The graph paper is very different for Polar coordinates.
To draw a graph in Polar coordinates, we need 2 values (for some graphs we will need 3 values), the length of the line ? and the angle of turn ? . The value of ? will be important in the diagrams that we draw.
These graphs are also called Roses and as such they have petals. We need to look at the relationship between ? ??? ? and the picture that we obtain.
Draw the following graphs ? ? = 4cos 3? ?? ? = 4cos 5? ??? ? = 4cos 7? ?? ? = 4cos ?? ? ? = 4cos(2?) ?? ? = 4cos(4?) ??? ? = 4cos(6?) ???? ? = 4cos ??
Draw the following graphs ? ? = 4sin 3? ?? ? = 4sin 5? ??? ? = 4sin 7? ?? ? = 4sin ?? ? ? = 4sin(2?) ?? ? = 4sin(4?) ??? ? = 4sin(6?) ???? ? = 4sin ??
With polar coordinates we can draw different types of graphs. ???? ??? (?????????) ???????????
??????? (?????????) We need to consider the following graphs ? = ? ?cos? ? = ? ?sin? We need to carefully look at the values of a and b .
Draw the following graphs ? ? = 1 + 2???? ?? ? = 1 2???? ??? ? = 4 + 4???? ?? ? = 5 + 3???? ? ? = 5 3????
For the graphs on the previous slide, change "cos" ??? "???"
??????????? With these curves, the value of the angle is always"2 " Draw the graph of ?2= ?2cos2?
?????? These are of the form ? 10 ? = ?
???????? Consider the equation 4? ? = 1 + ?cos?
With this equation we have 4 cases that we need to look at, ? ? = 0 ?? 0 < ? < 1 ??? ? = 1 ?? ? > 1
Converting between polar and Cartesian coordinates
To convert between polar and Cartesian coordinates we use the information from slide 3, which is ? = ?cos? ? = ?sin?
Find the cartesian equation of each of these curves ? = 4 ?cos? = 3 ?sin? = 7
Find the polar equation of each of these curves ?2+ ?2= 9 ?2 9+?2 16= 1 ?2+ ?2= 6?
Area enclosed by a polar curve. To do this we need to use the concept of differentiation that we first encounter in unit 1 of the A level Mathematics course.
