Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.4 Equations Involving Inverse Trigonometric Functions - 6.4 Exercises - Page 286: 33

$x=\frac{3}{5}$

Given: $arcsinx=arctan\frac{3}{4}$ Consider $arctan\frac{3}{4}=P$ $tan P=\frac{3}{4}$ Apply trigonometric identity for tangent. $tanx=\frac{opp}{adj}=\frac{3}{4}$ Thus, $hyp=\sqrt {3^{2}+4^{2}}$ Likewise, $sinP=\frac{opp}{hyp}=\frac{3}{5}$ $arcsinx=P$ gives $x=sinP$ Hence, $x=\frac{3}{5}$