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Let A and B denote the statements

A:\cos \alpha +\cos \beta +\cos \gamma = 0

B:\sin \alpha +\sin \beta +\sin \gamma = 0

If \cos \left (\beta -\gamma \right )+\cos \left ( \gamma -\alpha \right )+\cos \left ( \alpha -\beta \right )= -\frac{3}{2}, then

  • Option 1)

    A is false and B is true

  • Option 2)

    Both A and B are true

  • Option 3)

    Both A and B are false

  • Option 4)

    A is true and B is false

 

Answers (1)

best_answer

As we learnt in

 

Addition Formulae -

\cos \left ( A+B \right )= \cos A\cos B-\sin A\sin B

- wherein

A and B are two angles.

 

 \\ \cos (\beta - \gamma)+ \cos (\gamma - \alpha)+ \cos ( \alpha - \beta) = -\frac{3}{2} \\ \Rightarrow \cos\beta \ \cos \gamma+ \sin\beta \ \sin \gamma +\cos\gamma \ \cos \alpha + \sin\gamma \ \sin \alpha + \cos\alpha \ \cos \beta + \sin\alpha \ \sin \beta \\ = -\frac{3}{2}
\\ \Rightarrow 2 [ \cos\alpha \ \cos \beta+ \cos\beta \ \cos \gamma +\cos\gamma \ \cos \alpha] + 2 [ \sin \alpha \ \sin \beta+ \sin\beta \sin \gamma +\sin\gamma \ \sin \alpha] = -3\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, .................(1)

Adding  \\(\sin^{2}\alpha + \cos ^{2}\alpha), (\sin^{2}\beta + \cos ^{2}\beta), (\sin^{2}\gamma + \cos ^{2}\gamma) to equation, we get             

\\ (\sin^{2}\alpha+\sin^{2} \beta + \sin^{2} \gamma + 2 \sin \alpha \sin \beta+ 2\sin \beta \sin \gamma + 2 \sin \gamma \sin \alpha)+ (\cos^{2}\alpha + \cos ^{2}\beta + \cos^{2}\gamma +2 \cos\alpha\ cos\beta+ 2 \cos \beta \cos \gamma + 2 \cos \gamma \cos \alpha)

\\ = -3 + 1+1 + 1\, \, \, \, \, \, \, \, \, \, [ as\ sin 2x + \cos 2x =1]\\ =0

\Rightarrow \left [ \sin \alpha + \sin \beta + \sin \gamma \right ]^{2}+ \left [ \cos \alpha + \cos \beta + \cos \gamma \right ]^{2} = 0

This is only true if 

\\ \sin \alpha + \sin \beta + \sin \gamma \right ] = 0 \\ \cos \alpha + \cos \beta + \cos \gamma = 0

Hence both statements true.


Option 1)

A is false and B is true

This option is incorrect.

Option 2)

Both A and B are true

This option is correct.

Option 3)

Both A and B are false

This option is incorrect.

Option 4)

A is true and B is false

This option is incorrect.

Posted by

Plabita

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