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Popular >

solvefor x,z=sin(x^2y)

Solution

solve for

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

Solution

x = \frac{arcsin ( z ) + 2 πn}{y}, x = - \frac{arcsin ( z ) + 2 πn}{y}, x = \frac{π + arcsin ( - z ) + 2 πn}{y}, x = - \frac{π + arcsin ( - z ) + 2 πn}{y}

Solution steps

z = sin (x^{2} y)

Switch sides

sin (x^{2} y) = z

Apply trig inverse properties

sin (x^{2} y) = z

General solutions for

sin (x^{2} y) = z

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x^{2} y = arcsin (z) + 2 πn, x^{2} y = π + arcsin (- z) + 2 πn

x^{2} y = arcsin (z) + 2 πn, x^{2} y = π + arcsin (- z) + 2 πn

Solve

x^{2} y = arcsin (z) + 2 πn : x = \frac{arcsin ( z ) + 2 πn}{y}, x = - \frac{arcsin ( z ) + 2 πn}{y}; y \neq = 0

x^{2} y = arcsin (z) + 2 πn

Divide both sides by

y; y \neq = 0

x^{2} y = arcsin (z) + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x ^{2} y}{y} = \frac{arcsin ( z )}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

x^{2} = \frac{arcsin ( z )}{y} + \frac{2 πn}{y}; y \neq = 0

x^{2} = \frac{arcsin ( z )}{y} + \frac{2 πn}{y}; y \neq = 0

For

x^{2} = f (a)

the solutions are

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

x = \frac{arcsin ( z )}{y} + \frac{2 πn}{y}, x = - \frac{arcsin ( z )}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

\frac{arcsin ( z )}{y} + \frac{2 πn}{y} : \frac{arcsin ( z ) + 2 πn}{y}

\frac{arcsin ( z )}{y} + \frac{2 πn}{y}

Combine the fractions

\frac{arcsin ( z )}{y} + \frac{2 πn}{y} : \frac{arcsin ( z ) + 2 πn}{y}

Apply rule

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

= \frac{arcsin ( z ) + 2 πn}{y}

= \frac{arcsin ( z ) + 2 πn}{y}

Simplify

- \frac{arcsin ( z )}{y} + \frac{2 πn}{y} : - \frac{arcsin ( z ) + 2 πn}{y}

- \frac{arcsin ( z )}{y} + \frac{2 πn}{y}

Combine the fractions

\frac{arcsin ( z )}{y} + \frac{2 πn}{y} : \frac{arcsin ( z ) + 2 πn}{y}

Apply rule

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

= \frac{arcsin ( z ) + 2 πn}{y}

= - \frac{2 πn + arcsin ( z )}{y}

= - \frac{arcsin ( z ) + 2 πn}{y}

x = \frac{arcsin ( z ) + 2 πn}{y}, x = - \frac{arcsin ( z ) + 2 πn}{y}; y \neq = 0

Solve

x^{2} y = π + arcsin (- z) + 2 πn : x = \frac{π + arcsin ( - z ) + 2 πn}{y}, x = - \frac{π + arcsin ( - z ) + 2 πn}{y}; y \neq = 0

x^{2} y = π + arcsin (- z) + 2 πn

Divide both sides by

y; y \neq = 0

x^{2} y = π + arcsin (- z) + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x ^{2} y}{y} = \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

x^{2} = \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}; y \neq = 0

x^{2} = \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}; y \neq = 0

For

x^{2} = f (a)

the solutions are

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

x = \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}, x = - \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

\frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y} : \frac{π + arcsin ( - z ) + 2 πn}{y}

\frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}

Combine the fractions

\frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y} : \frac{π + arcsin ( - z ) + 2 πn}{y}

Apply rule

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

= \frac{π + arcsin ( - z ) + 2 πn}{y}

= \frac{π + arcsin ( - z ) + 2 πn}{y}

Simplify

- \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y} : - \frac{π + arcsin ( - z ) + 2 πn}{y}

- \frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y}

Combine the fractions

\frac{π}{y} + \frac{arcsin ( - z )}{y} + \frac{2 πn}{y} : \frac{π + arcsin ( - z ) + 2 πn}{y}

Apply rule

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

= \frac{π + arcsin ( - z ) + 2 πn}{y}

= - \frac{π + 2 πn + arcsin ( - z )}{y}

= - \frac{π + arcsin ( - z ) + 2 πn}{y}

x = \frac{π + arcsin ( - z ) + 2 πn}{y}, x = - \frac{π + arcsin ( - z ) + 2 πn}{y}; y \neq = 0

x = \frac{arcsin ( z ) + 2 πn}{y}, x = - \frac{arcsin ( z ) + 2 πn}{y}, x = \frac{π + arcsin ( - z ) + 2 πn}{y}, x = - \frac{π + arcsin ( - z ) + 2 πn}{y}

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