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Get Answers to all your Questions
Sin (x+y) = 3/5 , Cos (x-y) = 12/13 Find tan 2x ?
Answers (1)
Given- $\sin(x + y) = \frac{3}{5}$ and $\cos(x - y) = \frac{12}{13}$
To find - $\tan(2x)$
Solution- Using the identity: $\tan(2x) = \frac{\sin(2x)}{\cos(2x)}$.
Now use angle addition formulas:
$\sin(2x) = \sin(x + y)\cos(x - y) + \cos(x + y)\sin(x - y)$,
$\cos(2x) = \cos(x + y)\cos(x - y) - \sin(x + y)\sin(x - y)$.
Since $\sin(x + y) = \frac{3}{5}$, then $\cos(x + y) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}$.
Since $\cos(x - y) = \frac{12}{13}$, then $\sin(x - y) = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \frac{5}{13}$.
Now compute:
$\sin(2x) = \frac{3}{5} \cdot \frac{12}{13} + \frac{4}{5} \cdot \frac{5}{13} = \frac{36}{65} + \frac{20}{65} = \frac{56}{65}$.
$\cos(2x) = \frac{4}{5} \cdot \frac{12}{13} - \frac{3}{5} \cdot \frac{5}{13} = \frac{48}{65} - \frac{15}{65} = \frac{33}{65}$.
So, $\tan(2x) = \frac{56}{33}$.
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