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Write the Value of Sin a + Sin 3 a Cos a + Cos 3 a

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Question

Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]

Sum

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Solution

\[\frac{\sin A + \sin3A}{\cos A + \cos3A}\]
\[ = \frac{2\sin\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)}{2\cos\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)} \left[ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right), \text{ and }\cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = \frac{\sin2A \cos\left( - A \right)}{\cos2A \cos\left( - A \right)}\]
\[ = \frac{\sin2A \cos A}{\cos2A \cos A}\]
\[ =\tan2A\]

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Transformation Formulae

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Chapter 8: Transformation formulae - Exercise 8.3 [Page 21]

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R.D. Sharma Mathematics [English] Class 11

Chapter 8 Transformation formulae
Exercise 8.3 | Q 9 | Page 21

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