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Popular Trigonometry >

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

Solution

solve for

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

Solution

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

Solution steps

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

Switch sides

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

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

x^{2} + y^{2} = arcsin (z) + 2 πn

Move

y^{2}

to the right side

x^{2} + y^{2} = arcsin (z) + 2 πn

Subtract

y^{2}

from both sides

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

Simplify

x^{2} = arcsin (z) + 2 πn - y^{2}

x^{2} = arcsin (z) + 2 πn - y^{2}

For

x^{2} = f (a)

the solutions are

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

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

Solve

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

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

Move

y^{2}

to the right side

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

Subtract

y^{2}

from both sides

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

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

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

Graph

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