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

The general solution for 2csc^2(x)-sqrt(2)csc(x)-2=0 is x= pi/4+2pin,x=(3pi)/4+2pin

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2csc^2(x)-sqrt(2)csc(x)-2=0

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

csc (x) = u

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

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 2)^{2} - 4 \cdot 2 (- 2) = 32

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

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 2 \cdot 2

(- 2)^{2} = 2

(- 2)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= (2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

4 \cdot 2 \cdot 2 = 16

4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 16

= 2 + 16

Add the numbers:

2 + 16 = 18

= 18

Prime factorization of

18 : 2 \cdot 3^{2}

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

= 2 \cdot 3^{2}

= 3^{2} \cdot 2

Apply radical rule:

= 2 3^{2}

Apply radical rule:

3^{2} = 3

= 32

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

Separate the solutions

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

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

\frac{- ( - 2 ) + 3 2}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{2 + 3 2}{2 \cdot 2}

Add similar elements:

2 + 32 = 42

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 2}{4}

Divide the numbers:

\frac{4}{4} = 1

= 2

u = \frac{- ( - 2 ) - 3 2}{2 \cdot 2} : - \frac{2}{2}

\frac{- ( - 2 ) - 3 2}{2 \cdot 2}

Apply rule

- (- a) = a

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

Add similar elements:

2 - 32 = - 22

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2 2}{4}

Apply the fraction rule:

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

= - \frac{2 2}{4}

Cancel the common factor:

2

= - \frac{2}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = csc (x)

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

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

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 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{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Popular Examples

sin(t)=-(sqrt(2))/2

sin (t) = - \frac{2}{2}

9cos(2θ)=9cos^2(θ)-4

9 cos (2 θ) = 9 cos^{2} (θ) - 4

-10sin(x)=0

- 10 sin (x) = 0

sin(x/5)=0

sin (\frac{x}{5}) = 0

8400=624sin((2pi)/(365)t)+8736

8400 = 624 sin (\frac{2 π}{365} t) + 8736

Frequently Asked Questions (FAQ)

What is the general solution for 2csc^2(x)-sqrt(2)csc(x)-2=0 ?
The general solution for 2csc^2(x)-sqrt(2)csc(x)-2=0 is x= pi/4+2pin,x=(3pi)/4+2pin