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If a=b*cos (2pi)/3=c*cos (4pi)/3, then the value of (ab+bc+ca) is ?

If a=b*cos (2pi)/3=c*cos (4pi)/3, then the value of (ab+bc+ca) is ?

Grade:11

3 Answers

Swapnil Saxena
102 Points
9 years ago

Solution :

a=b*cos(2pi)/3 then (3*a)/cos(2pi)=b. Since cos(2pi)=1  ==>(3*a)=b

a=c*cos(4pi)/3 then (3*a)/cos(4pi)=c. Since cos(4pi)=1  ==>(3*a)=c

then (ab+bc+ca)=((3a)(a)+(3a)(3a)+(a)(3a))==>((3a^2)+(9a^2)+(3a^2))=(15*a^2) Ans

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Arnab,

 

cos(2pi/3) = cos[ pi - pi/3 ] = -cos(pi/3) = -1/2

And cos(4pi/3) = cos(pi + pi/3) = -cos(pi/3) = -1/2

 

So a = -b/2 = -c/2

So b = -2a, c = -2a

 

ab+bc+ca = a(-2a)+(-2a)(-2a)+(-2a)a = 4a^2 - 4a^2 = 0

 

Hence the expression is 0.

 

Hope that helps.

 

Best Regards,

Ashwin (IIT Madras).

Swapnil Saxena
102 Points
9 years ago

Sorry,

I wrongly interpreted th quetion as a=b*(cos (2pi))/3=c*(cos (4pi))/3, then the value of (ab+bc+ca).

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