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Sin 2 ( π 18 ) + Sin 2 ( π 9 ) + Sin 2 ( 7 π 18 ) + Sin 2 ( 4 π 9 ) = - Mathematics

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प्रश्न

\[\sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right) =\]

विकल्प

  • 1

  • 2

  • 4

  • none of these. 

MCQ
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उत्तर

2

\[\text{ We have, }  \]

\[ \sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right)\]

\[ = \frac{1}{2}\left[ 1 - \cos\left( \frac{\pi}{9} \right) + 1 - \cos\left( \frac{2\pi}{9} \right) + 1 - \cos\frac{7\pi}{9} + 1 - \cos\frac{8\pi}{9} \right] \left( \because \sin^2 \theta = \frac{1 - \cos2\theta}{2} \right)\]

\[ = \frac{1}{2}\left[ 4 - \cos\left( \frac{\pi}{9} \right) - \cos\left( \frac{2\pi}{9} \right) - \left\{ - \cos\left( \pi - \frac{7\pi}{9} \right) \right\} - \left\{ - \cos\left( \pi - \frac{8\pi}{9} \right) \right\} \right]\]

\[ = \frac{1}{2}\left[ 4 - \cos\left( \frac{\pi}{9} \right) - \cos\left( \frac{2\pi}{9} \right) + \cos\left( \frac{2\pi}{9} \right) + \cos\left( \frac{\pi}{9} \right) \right]\]

\[ = \frac{4}{2}\]

\[ = 2\]

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Values of Trigonometric Functions at Multiples and Submultiples of an Angle
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.5 [पृष्ठ ४४]

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आरडी शर्मा Mathematics [English] Class 11
अध्याय 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.5 | Q 14 | पृष्ठ ४४

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