What is the general solution for 2sin(x)+1/(sin(x))=0 ?

The general solution for 2sin(x)+1/(sin(x))=0 is No Solution for x\in\mathbb{R}

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2sin(x)+1/(sin(x))=0

Solution

2 sin (x) + \frac{1}{sin ( x )} = 0

Solution

No Solution for x \in R

Solution steps

2 sin (x) + \frac{1}{sin ( x )} = 0

Solve by substitution

2 sin (x) + \frac{1}{sin ( x )} = 0

Let:

sin (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

2 uu + \frac{1}{u} u = 0 \cdot u

Simplify

2 uu + \frac{1}{u} u = 0 \cdot u

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

2 u^{2} + 1 = 0

2 u^{2} + 1 = 0

2 u^{2} + 1 = 0

Solve

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

2 u^{2} + 1 = 0

Move

1

to the right side

2 u^{2} + 1 = 0

Subtract

1

from both sides

2 u^{2} + 1 - 1 = 0 - 1

Simplify

2 u^{2} = - 1

2 u^{2} = - 1

Divide both sides by

2

2 u^{2} = - 1

Divide both sides by

2

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

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

Simplify

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

- \frac{1}{2}

Apply radical rule:

- a = - 1 a

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

= - 1 \frac{1}{2}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{2}

Simplify

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

- - \frac{1}{2}

Simplify

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

- \frac{1}{2}

Apply radical rule:

- a = - 1 a

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

= - 1 \frac{1}{2}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{2}

= - i \frac{1}{2}

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

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

Substitute back

u = sin (x)

sin (x) = i \frac{1}{2}, sin (x) = - i \frac{1}{2}

sin (x) = i \frac{1}{2}, sin (x) = - i \frac{1}{2}

sin (x) = i \frac{1}{2} :

No Solution

sin (x) = i \frac{1}{2}

No Solution

sin (x) = - i \frac{1}{2} :

No Solution

sin (x) = - i \frac{1}{2}

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 2sin(x)+1/(sin(x))=0 ?
The general solution for 2sin(x)+1/(sin(x))=0 is No Solution for x\in\mathbb{R}