11th Class Mathematics Trigonometric Identities Question Bank Critical Thinking

  • question_answer \[\cos \,\,2\theta +2\,\,\cos \theta \] is always

    A) Greater than \[-\frac{3}{2}\]

    B) Less than or equal to \[\frac{3}{2}\]

    C) Greater than or equal to \[-\frac{3}{2}\] and less than or equal to 3

    D) None of these

    Correct Answer: C

    Solution :

    We have \[\cos 2\theta +2\cos \theta =2{{\cos }^{2}}\theta -1+2\cos \theta \]             \[=2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\] Now \[2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}\ge 0\]for all \[\theta \] \[\therefore \,\,2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\ge \frac{-3}{2}\]for all \[\theta \]. Þ \[\cos 2\theta +2\cos \theta \ge \frac{-3}{2}\]for all \[\theta \]

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