What is the general solution for sqrt(2)csc^2(x)+csc(x)=sqrt(2) ?

The general solution for sqrt(2)csc^2(x)+csc(x)=sqrt(2) is x=(5pi)/4+2pin,x=(7pi)/4+2pin

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sqrt(2)csc^2(x)+csc(x)=sqrt(2)

Solution

2 csc^{2} (x) + csc (x) = 2

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

2 csc^{2} (x) + csc (x) = 2

Solve by substitution

2 csc^{2} (x) + csc (x) = 2

Let:

csc (x) = u

2 u^{2} + u = 2

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

2 u^{2} + u = 2

Move

2

to the left side

2 u^{2} + u = 2

Subtract

2

from both sides

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

Simplify

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

1^{2} - 42 (- 2) = 3

1^{2} - 42 (- 2)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 42 (- 2)

Apply rule

- (- a) = a

= 1 + 422

422 = 8

422

Apply radical rule:

a a = a

22 = 2

= 4 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 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 2}

Separate the solutions

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

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

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

Add/Subtract the numbers:

- 1 + 3 = 2

= \frac{2}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= \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{- 1 - 3}{2 2} : - 2

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

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{2 2}

Apply the fraction rule:

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

= - \frac{4}{2 2}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = csc (x)

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

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

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) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

General solutions for

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)csc^2(x)+csc(x)=sqrt(2) ?
The general solution for sqrt(2)csc^2(x)+csc(x)=sqrt(2) is x=(5pi)/4+2pin,x=(7pi)/4+2pin