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solvefor x,z=arcsin((xy)/(x^2+y^2))

Solution

solve for

x, z = arcsin (\frac{x y}{x ^{2} + y ^{2}})

x = \frac{y + y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}, x = \frac{y - y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}

Solution steps

z = arcsin (\frac{x y}{x ^{2} + y ^{2}})

Switch sides

arcsin (\frac{x y}{x ^{2} + y ^{2}}) = z

Apply trig inverse properties

\frac{x y}{x ^{2} + y ^{2}} = sin (z)

Solve

\frac{x y}{x ^{2} + y ^{2}} = sin (z) : x = \frac{y + y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}, x = \frac{y - y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}; sin (z) \neq = 0

x = \frac{y + y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}, x = \frac{y - y 1 - 4 sin ^{2} ( z )}{2 sin ( z )}

\frac{tan ( x ) + cos ( x )}{( 1 + sin ( x ) )} = \frac{1}{cos ( x )}

sin (θ) = \frac{4}{5}, 0 \leq θ \leq \frac{π}{2}

- cos^{2} (x) + cos (x) + 2 = 0

2 - cos^{2} (x) = 4 sin^{2} (\frac{1}{2} x)

\frac{( tan ( x ) + 2 sin ( x ) )}{( tan ( x ) - 2 sin ( x ) )} = 3