11th Class Mathematics Trigonometric Identities Question Bank Critical Thinking

  • question_answer If \[\alpha ,\,\beta ,\,\gamma \in \,\left( 0,\,\frac{\pi }{2} \right)\], then \[\frac{\sin \,(\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }\] is  

    A) < 1

    B) >1

    C) = 1

    D) None of these

    Correct Answer: A

    Solution :

    We have \[\sin \alpha +\sin \beta +\sin \gamma -\sin (\alpha +\beta +\gamma )\] \[=\sin \alpha +\sin \beta +\sin \gamma -\sin \alpha \cos \beta \cos \gamma \] \[-\cos \alpha \sin \beta \cos \gamma -\cos \alpha \cos \beta \sin \gamma +\sin \alpha \sin \beta \sin \gamma \] \[=\sin \alpha (1-\cos \beta \cos \gamma )+\sin \beta (1-\cos \alpha \cos \gamma )\] \[+\sin \gamma (1-\cos \alpha \cos \beta )+\sin \alpha \sin \beta \sin \gamma >0\] \[\therefore \sin \alpha +\sin \beta +\sin \gamma >\sin (\alpha +\beta +\gamma )\] \[\Rightarrow \frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }<1\] .


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