What is the distance between the following polar coordinates?: <mstyle displaystyle="true">

hushjelpw4

hushjelpw4

Answered question

2022-05-30

What is the distance between the following polar coordinates?:
( 2 , 3 π 4 ) , ( 1 , 15 π 8 )

Answer & Explanation

Bruce Bridges

Bruce Bridges

Beginner2022-05-31Added 13 answers

Step 1
Distance between Polar Co-ordinates: A ( r 1 , θ 1 ) and B ( r 2 , θ 2 ) is
D = r 1 2 + r 2 2 - 2 r 1 r 2 cos ( θ 1 - θ 2 ) ... ( I )
We have, P 1 ( 2 , 3 π 4 ) and P 2 ( 1 , 15 π 8 )
So, r 1 = 2 , r 2 = 1 , θ 1 = 3 π 4 and θ 2 = 15 π 8
θ 1 - θ 2 = 3 π 4 - 15 π 8 = 6 π - 15 π 8 = - 9 π 8
Step 2
cos ( θ 1 - θ 2 ) = cos ( - 9 π 8 )
cos ( θ 1 - θ 2 ) = cos ( 9 π 8 ) [ cos ( - θ ) = cos θ ]
cos ( θ 1 - θ 2 ) = cos ( 9 π 8 ) [ cos ( - θ ) = cos θ ]
cos ( θ 1 - θ 2 ) = cos ( π + π 8 ) = - cos ( π 8 )
Using: (I) we get
D = 2 2 + 1 2 - 2 ( 2 ) ( 1 ) ( - cos ( π 8 ) )
D = 4 + 1 + 4 cos ( π 8 )
D = 5 + 4 cos ( π 8 )
D 2.9488
Jude Hunt

Jude Hunt

Beginner2022-06-01Added 4 answers

Step 1
It is nontrivial to determine polar distances (since drawing a line in polar coordinates isn't very easy), so we should convert to Cartesian coordinates.
The first point is moderately easy:
x = r cos θ = 2 - 2 2 = - 2
y = r sin θ = 2 2 2 = 2
The second point is a little harder, but we only need to use half angle identities. Let
θ = 15 π 4 - π 4 . Therefore, cos ( θ ) = 2 2 and
cos ( θ 2 ) = ± 1 + cos θ 2 = ± 2 + 2 2
sin ( θ 2 ) = ± 1 - cos θ 2 = ± 2 - 2 2
Since 15 π 8 s in the 4th quadrant, its cosine is positive and its sine is negative. Therefore,
x = r cos ( 15 π 8 ) = 2 + 2 2
y = r sin ( 15 π 8 ) = 2 - 2 2
This allows us to use Pythagorean theorem in order to find the distance between the two points:
Δ x = 2 + 2 2 + 2
Δ y = 2 - 2 2 - 2
Pulling out a factor of 2 from each, then squaring and adding them,
d s 2 = 2 [ 1 + 2 2 + 1 + 1 - 2 2 + 1 + 2 1 + 2 2 - 2 1 - 2 2 ]
= 2 [ 4 + 2 ( 1 + 2 2 - 1 - 2 2 ) ]
Using the cosine sum formula with π 8 and π 4 , we get
= 8 ( 1 + cos ( 3 π 8 ) ) = 4 cos 2 ( 3 π 16 )
Therefore the total distance is
D = 2 cos ( 3 π 16 ) = 2 + 2 - 2 1.66

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