What is the general solution for (sin(x))/1+4/(sin(x))+5=0 ?

The general solution for (sin(x))/1+4/(sin(x))+5=0 is x=(3pi)/2+2pin

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(sin(x))/1+4/(sin(x))+5=0

Solution

\frac{sin ( x )}{1} + \frac{4}{sin ( x )} + 5 = 0

Solution

x = \frac{3 π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sin ( x )}{1} + \frac{4}{sin ( x )} + 5 = 0

Solve by substitution

\frac{sin ( x )}{1} + \frac{4}{sin ( x )} + 5 = 0

Let:

sin (x) = u

\frac{u}{1} + \frac{4}{u} + 5 = 0

\frac{u}{1} + \frac{4}{u} + 5 = 0 : u = - 1, u = - 4

\frac{u}{1} + \frac{4}{u} + 5 = 0

Simplify

\frac{u}{1} : u

\frac{u}{1}

Apply rule

\frac{a}{1} = a

= u

u + \frac{4}{u} + 5 = 0

Multiply both sides by

u

u + \frac{4}{u} + 5 = 0

Multiply both sides by

u

uu + \frac{4}{u} u + 5 u = 0 \cdot u

Simplify

uu + \frac{4}{u} u + 5 u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{4}{u} u : 4

\frac{4}{u} u

Multiply fractions:

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

= \frac{4 u}{u}

Cancel the common factor:

u

= 4

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 4 + 5 u = 0

u^{2} + 4 + 5 u = 0

u^{2} + 4 + 5 u = 0

Solve

u^{2} + 4 + 5 u = 0 : u = - 1, u = - 4

u^{2} + 4 + 5 u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} + 5 u + 4 = 0

Solve with the quadratic formula

u^{2} + 5 u + 4 = 0

Quadratic Equation Formula:

For

a = 1, b = 5, c = 4

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

5^{2} - 4 \cdot 1 \cdot 4 = 3

5^{2} - 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Subtract the numbers:

25 - 16 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 5 \pm 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 5 + 3}{2 \cdot 1}, u_{2} = \frac{- 5 - 3}{2 \cdot 1}

u = \frac{- 5 + 3}{2 \cdot 1} : - 1

\frac{- 5 + 3}{2 \cdot 1}

Add/Subtract the numbers:

- 5 + 3 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- 5 - 3}{2 \cdot 1} : - 4

\frac{- 5 - 3}{2 \cdot 1}

Subtract the numbers:

- 5 - 3 = - 8

= \frac{- 8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8}{2}

Apply the fraction rule:

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

= - \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= - 4

The solutions to the quadratic equation are:

u = - 1, u = - 4

u = - 1, u = - 4

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{u}{1} + \frac{4}{u} + 5

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1, u = - 4

Substitute back

u = sin (x)

sin (x) = - 1, sin (x) = - 4

sin (x) = - 1, sin (x) = - 4

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = - 4 :

No Solution

sin (x) = - 4

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))/1+4/(sin(x))+5=0 ?
The general solution for (sin(x))/1+4/(sin(x))+5=0 is x=(3pi)/2+2pin