What is the distance between $(2 ,(7 pi)/4 )$ and $(5 , (11 pi )/12 )$ ?

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Answer 1 · 2018-07-04T07:20:32

$D=sqrt(29+10sqrt3)$

$orD~~6.81$

We know that ,

$"Distance between two polar co-ordinates "$

$A(r_1,theta_1) and B(r_2,theta_2)$ is :

$color(red)(D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))...to(1)$

We have , $A(2,(7pi)/4) and B(5,(11pi)/12)$

$:.r_1=2 ,r_2=5,theta_1=(7pi)/4 and theta_2=(11pi)/12$

$:.cos(theta_1-theta_2)=cos((7pi)/4-(11pi)/12)=cos((21pi-11pi)/12)$

$=>cos(theta_1-theta_2)=((-10pi)/12)=cos(-(5pi)/6)$

$=>cos(theta_1-theta_2)=cos((5pi)/6)...to[becausecos(-theta)=costheta]$

$cos(theta_1-theta_2)=cos(pi-pi/6)=-cos(pi/6)to[becauseII^(nd)Quad.]$

$=>cos(theta_1-theta_2)=-sqrt3/2$

So ,from $(1)$

$D=sqrt(2^2+5^2-2(2)(5)(-sqrt3/2))$

$=>D=sqrt(4+25+20(sqrt3/2))$

$=>D=sqrt(29+10sqrt3)$

$=>D~~6.81$

What is the distance between (2 ,(7 pi)/4 )(2 ,(7 pi)/4 ) and (5 , (11 pi )/12 )(5 , (11 pi )/12 )?