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

Raphael Mccullough 2022-04-12 Answered
What is the distance between the following polar coordinates?: ( 1 , 3 π 4 ) , ( 5 , 5 π 8 )
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Answers (1)

Madalynn Acosta
Answered 2022-04-13 Author has 15 answers
Step 1
Call the two points A and B:
A = ( r a , θ a ) = ( 1 , 3 π 4 )
B = ( r b , θ b ) = ( 5 , 5 π 8 )
Convert them to rectangular form:
A = ( x a , y a )
x a = r a cos ( θ a ) = 1 cos ( 3 π 4 ) = - 2 2
y a = r a sin ( θ a ) = 1 sin ( 3 π 4 ) = 2 2
A = ( - 2 2 , 2 2 ) ( - 0.707 , 0.707 )
B = ( x b , y b )
x b = r b cos ( θ b ) = 5 cos ( 5 π 8 ) = - 5 2 - 2 2
y b = r b sin ( θ b ) = 5 sin ( 5 π 8 ) = 5 2 + 2 2
B = ( - 5 2 - 2 2 , 5 2 + 2 2 ) ( - 1.913 , 4.619 )
Now apply the Pythagorean Theorem to find the length of the line segment A B ¯
| | A B ¯ | | = ( x b - x a ) 2 + ( y b - y a ) 2
| | A B ¯ | | ( ( - 1.913 ) - ( - 0.707 ) ) 2 + ( 4.619 - 0.707 ) 2
| | A B ¯ | | ( - 1.206 ) 2 + ( 3.912 ) 2 = 1.455 + 15.306
| | A B ¯ | | 16.761 4.094
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