A) 3/2
B) - 3/2
C) 0
D) 1
Correct Answer: D
Solution :
\[\frac{b}{c}=\frac{\cos \beta +i\sin \beta }{\cos \gamma +i\sin \gamma }\times \frac{\cos \gamma -i\sin \gamma }{\cos \gamma -i\sin \gamma }\] \[\Rightarrow \frac{b}{c}=\,\cos (\beta -\gamma )+i\sin (\beta -\gamma )\] ......(i) Similarly,\[\,\frac{c}{a}=\cos (\gamma -\alpha )+i\sin \,(\gamma -\alpha )\] ......(ii) and \[\,\,\frac{a}{b}=\cos (\alpha -\beta )+i\sin (\alpha -\beta )\] .....(iii) from (i) + (ii) + (iii) \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )\]\[+i[\sin (\beta -\gamma )+\sin (\gamma -\alpha )+\sin (\alpha -\beta )]=1\] Equating real and imaginary parts, \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=1\].
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