What is the general solution for -csc(x)+2csc^3(x)=0 ?

The general solution for -csc(x)+2csc^3(x)=0 is No Solution for x\in\mathbb{R}

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Popular >

-csc(x)+2csc^3(x)=0

Solution

- csc (x) + 2 csc^{3} (x) = 0

Solution

No Solution for x \in R

Solution steps

- csc (x) + 2 csc^{3} (x) = 0

Solve by substitution

- csc (x) + 2 csc^{3} (x) = 0

Let:

csc (x) = u

- u + 2 u^{3} = 0

- u + 2 u^{3} = 0 : u = 0, u = - \frac{2}{2}, u = \frac{2}{2}

- u + 2 u^{3} = 0

Factor

- u + 2 u^{3} : u (2 u + 1) (2 u - 1)

- u + 2 u^{3}

Factor out common term

u : u (2 u^{2} - 1)

2 u^{3} - u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= 2 u^{2} u - u

Factor out common term

u

= u (2 u^{2} - 1)

= u (2 u^{2} - 1)

Factor

2 u^{2} - 1 : (2 u + 1) (2 u - 1)

2 u^{2} - 1

Rewrite

2 u^{2} - 1

(2 u)^{2} - 1^{2}

2 u^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} u^{2} - 1

Rewrite

1

1^{2}

= (2)^{2} u^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - 1^{2}

= (2 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(2 u)^{2} - 1^{2} = (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u - 1)

= u (2 u + 1) (2 u - 1)

u (2 u + 1) (2 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 2 u + 1 = 0 or 2 u - 1 = 0

Solve

2 u + 1 = 0 : u = - \frac{2}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

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

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

Solve

2 u - 1 = 0 : u = \frac{2}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

The solutions are

u = 0, u = - \frac{2}{2}, u = \frac{2}{2}

Substitute back

u = csc (x)

csc (x) = 0, csc (x) = - \frac{2}{2}, csc (x) = \frac{2}{2}

csc (x) = 0, csc (x) = - \frac{2}{2}, csc (x) = \frac{2}{2}

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

csc (x) = - \frac{2}{2} :

No Solution

csc (x) = - \frac{2}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

csc (x) = \frac{2}{2} :

No Solution

csc (x) = \frac{2}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for -csc(x)+2csc^3(x)=0 ?
The general solution for -csc(x)+2csc^3(x)=0 is No Solution for x\in\mathbb{R}