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solvefor x,z=sqrt(y)sin(4x)

Solution

solve for

x, z = y sin (4 x)

Solution

x = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{πn}{2}, x = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{πn}{2}

Solution steps

z = y sin (4 x)

Switch sides

y sin (4 x) = z

Divide both sides by

y; y \neq = 0

y sin (4 x) = z

Divide both sides by

y; y \neq = 0

\frac{y sin ( 4 x )}{y} = \frac{z}{y}; y \neq = 0

Simplify

\frac{y sin ( 4 x )}{y} = \frac{z}{y}

Simplify

\frac{y sin ( 4 x )}{y} : sin (4 x)

\frac{y sin ( 4 x )}{y}

Cancel the common factor:

y

= sin (4 x)

Simplify

\frac{z}{y} : \frac{z y}{y}

\frac{z}{y}

Multiply by the conjugate

\frac{y}{y}

= \frac{z y}{y y}

y y = y

y y

Apply radical rule:

a a = a

y y = y

= y

= \frac{z y}{y}

sin (4 x) = \frac{z y}{y}; y \neq = 0

sin (4 x) = \frac{z y}{y}; y \neq = 0

sin (4 x) = \frac{z y}{y}; y \neq = 0

Apply trig inverse properties

sin (4 x) = \frac{z y}{y}

General solutions for

sin (4 x) = \frac{z y}{y}

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

4 x = arcsin (\frac{z y}{y}) + 2 πn, 4 x = π + arcsin (- \frac{z y}{y}) + 2 πn

4 x = arcsin (\frac{z y}{y}) + 2 πn, 4 x = π + arcsin (- \frac{z y}{y}) + 2 πn

Solve

4 x = arcsin (\frac{z y}{y}) + 2 πn : x = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{πn}{2}

4 x = arcsin (\frac{z y}{y}) + 2 πn

Simplify

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

arcsin (\frac{z y}{y}) + 2 πn

\frac{z y}{y} = \frac{z}{y}

\frac{z y}{y}

Apply radical rule:

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

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

= \frac{z y ^{\frac{1}{2}}}{y}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{y ^{\frac{1}{2}}}{y ^{1}} = \frac{1}{y ^{1 - \frac{1}{2}}}

= \frac{z}{y ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{z}{y ^{\frac{1}{2}}}

Apply radical rule:

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

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

= \frac{z}{y}

= arcsin (\frac{z}{y}) + 2 πn

4 x = arcsin (\frac{z}{y}) + 2 πn

Divide both sides by

4

4 x = arcsin (\frac{z}{y}) + 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{2 πn}{4}

Simplify

x = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{πn}{2}

x = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{πn}{2}

Solve

4 x = π + arcsin (- \frac{z y}{y}) + 2 πn : x = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{πn}{2}

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

Simplify

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

π + arcsin (- \frac{z y}{y}) + 2 πn

\frac{z y}{y} = \frac{z}{y}

\frac{z y}{y}

Apply radical rule:

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

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

= \frac{z y ^{\frac{1}{2}}}{y}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{y ^{\frac{1}{2}}}{y ^{1}} = \frac{1}{y ^{1 - \frac{1}{2}}}

= \frac{z}{y ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{z}{y ^{\frac{1}{2}}}

Apply radical rule:

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

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

= \frac{z}{y}

= π + arcsin (- \frac{z}{y}) + 2 πn

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

Divide both sides by

4

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

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{2 πn}{4}

Simplify

x = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{πn}{2}

x = \frac{arcsin ( \frac{z}{y} )}{4} + \frac{πn}{2}, x = \frac{π}{4} + \frac{arcsin ( - \frac{z}{y} )}{4} + \frac{πn}{2}

Graph

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