# How do you find the exact value of Sin(arcsin(4/5)+arctan(12/5))?

Nov 8, 2016

$S \in \left(\arcsin \left(\frac{4}{5}\right) + \arctan \left(\frac{12}{5}\right)\right)$

$= S \in \left(\arcsin \left(\frac{4}{5}\right) + a r c \cot \left(\frac{5}{12}\right)\right)$

=Sin(arcsin(4/5)+arc csc(sqrt(1+(5/12)^2)

$= S \in \left(\arcsin \left(\frac{4}{5}\right) + a r c \csc \left(\frac{13}{12}\right)\right)$

$= S \in \left(\arcsin \left(\frac{4}{5}\right) + a r c \sin \left(\frac{12}{13}\right)\right)$

=Sin(arcsin((4/5)xxsqrt(1-(12/13)^2)+(12/13)xxsqrt(1-(4/5)^2))

$= \left(\frac{4}{5} \times \frac{5}{13}\right) + \left(\frac{12}{13} \times \frac{3}{5}\right) = \frac{56}{65}$