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

Solution

solve for

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

Solution

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

Solution steps

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

Switch sides

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

Apply trig inverse properties

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

arcsin (x) = a \Rightarrow x = sin (a)

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

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

Solve

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

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

Multiply both sides by

x^{2} + y^{2}

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

Simplify

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

Square both sides:

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

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

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

Expand

(sin (z) x^{2} + y^{2})^{2} : x^{2} sin^{2} (z) + y^{2} sin^{2} (z)

(sin (z) x^{2} + y^{2})^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

(x^{2} + y^{2})^{2} : x^{2} + y^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((x^{2} + y^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (x^{2} + y^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= x^{2} + y^{2}

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

Expand

(x^{2} + y^{2}) sin^{2} (z) : x^{2} sin^{2} (z) + y^{2} sin^{2} (z)

(x^{2} + y^{2}) sin^{2} (z)

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

Apply the distributive law:

a (b + c) = ab + a c

a = sin^{2} (z), b = x^{2}, c = y^{2}

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

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

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

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

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

Solve

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

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

Move

x^{2} sin^{2} (z)

to the left side

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

Subtract

x^{2} sin^{2} (z)

from both sides

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

Simplify

x^{2} - x^{2} sin^{2} (z) = y^{2} sin^{2} (z)

x^{2} - x^{2} sin^{2} (z) = y^{2} sin^{2} (z)

Factor

x^{2} - x^{2} sin^{2} (z) : x^{2} (1 - sin^{2} (z))

x^{2} - x^{2} sin^{2} (z)

Factor out common term

x^{2}

= x^{2} (1 - sin^{2} (z))

x^{2} (1 - sin^{2} (z)) = y^{2} sin^{2} (z)

Divide both sides by

1 - sin^{2} (z)

x^{2} (1 - sin^{2} (z)) = y^{2} sin^{2} (z)

Divide both sides by

1 - sin^{2} (z)

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

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

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

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

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