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If Sin a = 4 5 and Cos B = 5 13 , Where 0 < A, B < π 2 , Find the Value of the Following:Sin (A + B) - CBSE (Science) Class 11 - Mathematics

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Question

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:

sin (A + B)

 

Solution

Given: 
\[ \sin A = \frac{4}{5}\text{ and }\cos B = \frac{5}{13}\]
We know that
\[ \cos A = \sqrt{1 - \sin^2 A}\text{ and }\sin B = \sqrt{1 - \cos^2 B} ,\text{ where }0 < A , B < \frac{\pi}{2}\]
\[ \Rightarrow \cos A = \sqrt{1 - \left( \frac{4}{5} \right)^2} \text{ and }\sin B = \sqrt{1 - \left( \frac{5}{13} \right)^2}\]
\[ \Rightarrow \cos A = \sqrt{1 - \frac{16}{25}}\text{ and }\sin B = \sqrt{1 - \frac{25}{169}}\]
\[ \Rightarrow \cos A = \sqrt{\frac{9}{25}}\text{ and }\sin B = \sqrt{\frac{144}{169}}\]
\[ \Rightarrow \cos A = \frac{3}{5}\text{ and }\sin B = \frac{12}{13}\]
Now,
\[ \sin\left( A + B \right) = \sin A \cos B + \cos A \sin B\]
\[ = \frac{4}{5} \times \frac{5}{13} + \frac{3}{5} \times \frac{12}{13}\]
\[ = \frac{20}{65} + \frac{36}{65}\]
\[ = \frac{56}{65}\]

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APPEARS IN

 RD Sharma Solution for Mathematics Class 11 (2019 to Current)
Chapter 7: Values of Trigonometric function at sum or difference of angles
Ex.7.10 | Q: 1.1 | Page no. 19
Solution If Sin a = 4 5 and Cos B = 5 13 , Where 0 < A, B < π 2 , Find the Value of the Following:Sin (A + B) Concept: Trigonometric Functions of Sum and Difference of Two Angles.
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