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#