Question

Assertion :If $cos(\beta -\gamma )+cos(\gamma -\alpha )+cos(\alpha -\beta )=-\frac{3}{2}$, then Reason: $cos\alpha +cos\beta +cos\gamma =0$ and $sin\alpha +sin\beta +sin\gamma =0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$cos(\beta -r)+cos(r-\alpha )+cos(\alpha +\beta )=-3/2$

$\Rightarrow cos\beta cosr+sin\beta sinr+cosrcos\alpha +sinr+sin\alpha +cos\alpha cos\beta +sin\alpha sin\beta =-3/2.$

$\Rightarrow 2cos\beta cosr+2sin\beta sinr+2cosrcos\alpha +2sin\alpha rsin\alpha +cos\alpha cos\beta +2sin\alpha sin\beta =-3$

$\Rightarrow (cos\alpha +cos\beta +cosr{)}^2-co{s}^2\alpha -co{s}^2\beta -co{s}^{2}r+(sin\alpha +sin\beta +sinr{)}^2-si{n}^2-si{n}^2\alpha -si{n}^2\beta -si{n}^{2}r=-3$

$\Rightarrow (cos\alpha +cos\beta {)}^2-co{s}^2\alpha -co{s}^2\beta -co{s}^{2}r+(sin\alpha +sin\beta +sinr{)}^2-si{n}^2\alpha -si{n}^2\beta -si{n}^{2}r=-3$

$\Rightarrow (cos\alpha +cos\beta {)}^2+(sin\alpha +sin\beta +sinr{)}^2-(co{s}^2\alpha +si{n}^2\alpha )-(co{s}^2\beta +si{n}^2\beta )-(co{s}^{2}r+si{n}^{2}r)=-3$

$\Rightarrow (cos\alpha +cos\beta +cosr{)}^2+(sin\alpha +sin\beta +sinr{)}^2-1-1-1+3=0$

$\Rightarrow (cos\alpha +cos\beta +cosr{)}^2+(sin\alpha +sin\beta +sinr{)}^2=0$

$\therefore cos\alpha +cos\beta +cosr=0andsin\alpha +sin\beta +sinr=0$

thre both co statement ie A and B are true.