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Popular >

solvefor x,y^3-sin^2(x)=1

Solution

solve for

x, y^{3} - sin^{2} (x) = 1

Solution

x = arcsin (- 1 + y^{3}) + 2 πn, x = π + arcsin (- - 1 + y^{3}) + 2 πn, x = arcsin (- - 1 + y^{3}) + 2 πn, x = π + arcsin (- 1 + y^{3}) + 2 πn

Solution steps

y^{3} - sin^{2} (x) = 1

Solve by substitution

y^{3} - sin^{2} (x) = 1

Let:

sin (x) = u

y^{3} - u^{2} = 1

y^{3} - u^{2} = 1 : u = - 1 + y^{3}, u = - - 1 + y^{3}

y^{3} - u^{2} = 1

Move

y^{3}

to the right side

y^{3} - u^{2} = 1

Subtract

y^{3}

from both sides

y^{3} - u^{2} - y^{3} = 1 - y^{3}

Simplify

- u^{2} = 1 - y^{3}

- u^{2} = 1 - y^{3}

Divide both sides by

- 1

- u^{2} = 1 - y^{3}

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{1}{- 1} - \frac{y ^{3}}{- 1}

Simplify

\frac{- u ^{2}}{- 1} = \frac{1}{- 1} - \frac{y ^{3}}{- 1}

Simplify

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Apply rule

\frac{a}{1} = a

= u^{2}

Simplify

\frac{1}{- 1} - \frac{y ^{3}}{- 1} : - 1 + y^{3}

\frac{1}{- 1} - \frac{y ^{3}}{- 1}

Apply rule

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

= \frac{1 - y ^{3}}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1 - y ^{3}}{1}

Apply rule

\frac{a}{1} = a

= - (- y^{3} + 1)

Distribute parentheses

= - 1 - (- y^{3})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + y^{3}

u^{2} = - 1 + y^{3}

u^{2} = - 1 + y^{3}

u^{2} = - 1 + y^{3}

For

x^{2} = f (a)

the solutions are

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

u = - 1 + y^{3}, u = - - 1 + y^{3}

Substitute back

u = sin (x)

sin (x) = - 1 + y^{3}, sin (x) = - - 1 + y^{3}

sin (x) = - 1 + y^{3}, sin (x) = - - 1 + y^{3}

sin (x) = - 1 + y^{3} : x = arcsin (- 1 + y^{3}) + 2 πn, x = π + arcsin (- - 1 + y^{3}) + 2 πn

sin (x) = - 1 + y^{3}

Apply trig inverse properties

sin (x) = - 1 + y^{3}

General solutions for

sin (x) = - 1 + y^{3}

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

x = arcsin (- 1 + y^{3}) + 2 πn, x = π + arcsin (- - 1 + y^{3}) + 2 πn

x = arcsin (- 1 + y^{3}) + 2 πn, x = π + arcsin (- - 1 + y^{3}) + 2 πn

sin (x) = - - 1 + y^{3} : x = arcsin (- - 1 + y^{3}) + 2 πn, x = π + arcsin (- 1 + y^{3}) + 2 πn

sin (x) = - - 1 + y^{3}

Apply trig inverse properties

sin (x) = - - 1 + y^{3}

General solutions for

sin (x) = - - 1 + y^{3}

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

x = arcsin (- - 1 + y^{3}) + 2 πn, x = π + arcsin (- 1 + y^{3}) + 2 πn

x = arcsin (- - 1 + y^{3}) + 2 πn, x = π + arcsin (- 1 + y^{3}) + 2 πn

Combine all the solutions

x = arcsin (- 1 + y^{3}) + 2 πn, x = π + arcsin (- - 1 + y^{3}) + 2 πn, x = arcsin (- - 1 + y^{3}) + 2 πn, x = π + arcsin (- 1 + y^{3}) + 2 πn

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