English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

2csc^2(2x)-csc(2x)-6=0

Solution

2 csc^{2} (2 x) - csc (2 x) - 6 = 0

Solution

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.72972 \dots}{2} + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n, x = - 20.90515 \dots^{\circ} + 18 0^{\circ} n, x = 110.90515 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 csc^{2} (2 x) - csc (2 x) - 6 = 0

Solve by substitution

2 csc^{2} (2 x) - csc (2 x) - 6 = 0

Let:

csc (2 x) = u

2 u^{2} - u - 6 = 0

2 u^{2} - u - 6 = 0 : u = 2, u = - \frac{3}{2}

2 u^{2} - u - 6 = 0

Solve with the quadratic formula

2 u^{2} - u - 6 = 0

Quadratic Equation Formula:

For

a = 2, b = - 1, c = - 6

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

(- 1)^{2} - 4 \cdot 2 (- 6) = 7

(- 1)^{2} - 4 \cdot 2 (- 6)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 2 \cdot 6

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 6 = 48

4 \cdot 2 \cdot 6

Multiply the numbers:

4 \cdot 2 \cdot 6 = 48

= 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 2}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 2}

u = \frac{- ( - 1 ) + 7}{2 \cdot 2} : 2

\frac{- ( - 1 ) + 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 + 7}{2 \cdot 2}

Add the numbers:

1 + 7 = 8

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 1 ) - 7}{2 \cdot 2} : - \frac{3}{2}

\frac{- ( - 1 ) - 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{1 - 7}{2 \cdot 2}

Subtract the numbers:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 6}{4}

Apply the fraction rule:

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

= - \frac{6}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = 2, u = - \frac{3}{2}

Substitute back

u = csc (2 x)

csc (2 x) = 2, csc (2 x) = - \frac{3}{2}

csc (2 x) = 2, csc (2 x) = - \frac{3}{2}

csc (2 x) = 2 : x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

csc (2 x) = 2

General solutions for

csc (2 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

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

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

csc (2 x) = - \frac{3}{2}

Apply trig inverse properties

csc (2 x) = - \frac{3}{2}

General solutions for

csc (2 x) = - \frac{3}{2}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

2 x = arccsc (- \frac{3}{2}) + 2 πn, 2 x = π + arccsc (\frac{3}{2}) + 2 πn

2 x = arccsc (- \frac{3}{2}) + 2 πn, 2 x = π + arccsc (\frac{3}{2}) + 2 πn

Solve

2 x = arccsc (- \frac{3}{2}) + 2 πn : x = - \frac{arccsc ( \frac{3}{2} )}{2} + πn

2 x = arccsc (- \frac{3}{2}) + 2 πn

Simplify

arccsc (- \frac{3}{2}) + 2 πn : - arccsc (\frac{3}{2}) + 2 πn

arccsc (- \frac{3}{2}) + 2 πn

Use the following property:

arccsc (- x) = - arccsc (x)

arccsc (- \frac{3}{2}) = - arccsc (\frac{3}{2})

= - arccsc (\frac{3}{2}) + 2 πn

2 x = - arccsc (\frac{3}{2}) + 2 πn

Divide both sides by

2

2 x = - arccsc (\frac{3}{2}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arccsc ( \frac{3}{2} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arccsc ( \frac{3}{2} )}{2} + πn

x = - \frac{arccsc ( \frac{3}{2} )}{2} + πn

Solve

2 x = π + arccsc (\frac{3}{2}) + 2 πn : x = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + πn

2 x = π + arccsc (\frac{3}{2}) + 2 πn

Divide both sides by

2

2 x = π + arccsc (\frac{3}{2}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + πn

x = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + πn

x = - \frac{arccsc ( \frac{3}{2} )}{2} + πn, x = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + πn

Combine all the solutions

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{arccsc ( \frac{3}{2} )}{2} + πn, x = \frac{π}{2} + \frac{arccsc ( \frac{3}{2} )}{2} + πn

Show solutions in decimal form

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.72972 \dots}{2} + πn

Graph

Popular Examples

sin(2x)cos(x)=6sin^3(x)

cos(x)=-0.564

6sin(θ/2)=6cos(θ/2)

2cos(x/2)+sqrt(2)=0

sin(2x)+2cos(x)=0,0<= x<= 2pi