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If Sin X = 4 5 and 0 < X < π 2 , Find the Value of Sin 4x.

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प्रश्न

 If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]

, find the value of sin 4x.

 

 

संख्यात्मक
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उत्तर

\[\sin x = \frac{4}{5}\] and  \[0 < x < \frac{\pi}{2}\] .
\[\therefore \sin x = \sqrt{1 - \cos^2 x}\]
\[ \Rightarrow \left( \frac{4}{5} \right)^2 = 1 - \cos^2 x\]
\[ \Rightarrow \frac{16}{25} - 1 = - \cos^2 x\]
\[ \Rightarrow \frac{9}{25} = \cos^2 x\]
\[ \Rightarrow \cos x = \pm \frac{3}{5}\]
Since x lies in the 1st quadrant, cos x is positive.
Thus,
\[\cos x = \frac{3}{5}\]
Now,
\[\sin \left( 4x \right) = 2 \sin \left( 2x \right) \cos\left( 2x \right)\]
\[ = 2\left( 2 \sin x \cos x \right)\left( 1 - 2 \sin^2 x \right)\]
\[ = 2\left( 2 \times \frac{4}{5} \times \frac{3}{5} \right)\left( 1 - 2 \left( \frac{4}{5} \right)^2 \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( 1 - \frac{32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{25 - 32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{- 7}{25} \right)\]
\[ = - \frac{336}{625}\]
Hence, the value of sin 4x is \[- \frac{336}{625}\] . 


 

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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 9.1 [पृष्ठ २९]

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आर.डी. शर्मा Mathematics [English] Class 11
अध्याय 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 30.3 | पृष्ठ २९

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