What is the general solution for 1=(2-sin(x))/(sin^2(x)) ?

The general solution for 1=(2-sin(x))/(sin^2(x)) is x= pi/2+2pin

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1=(2-sin(x))/(sin^2(x))

Solution

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

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Switch sides

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Multiply both sides by

u^{2}

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

Multiply both sides by

u^{2}

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

Simplify

2 - u = u^{2}

2 - u = u^{2}

Solve

2 - u = u^{2} : u = 1, u = - 2

2 - u = u^{2}

Switch sides

u^{2} = 2 - u

Move

u

to the left side

u^{2} = 2 - u

Add

u

to both sides

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

Simplify

u^{2} + u = 2

u^{2} + u = 2

Move

2

to the left side

u^{2} + u = 2

Subtract

2

from both sides

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

Simplify

u^{2} + u - 2 = 0

u^{2} + u - 2 = 0

Solve with the quadratic formula

u^{2} + u - 2 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 2

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

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

1^{2} - 4 \cdot 1 \cdot (- 2) = 3

1^{2} - 4 \cdot 1 \cdot (- 2)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 2)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

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

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

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

Add/Subtract the numbers:

- 1 + 3 = 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

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

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

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4}{2}

Apply the fraction rule:

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

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

The solutions to the quadratic equation are:

u = 1, u = - 2

u = 1, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 2

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = - 2

sin (x) = 1, sin (x) = - 2

sin (x) = 1 : x = \frac{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{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 1=(2-sin(x))/(sin^2(x)) ?
The general solution for 1=(2-sin(x))/(sin^2(x)) is x= pi/2+2pin