What is the general solution for 4csc^2(x/2)-6csc(x/2)-4=0 ?

The general solution for 4csc^2(x/2)-6csc(x/2)-4=0 is x= pi/3+4pin,x=(5pi)/3+4pin

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4csc^2(x/2)-6csc(x/2)-4=0

Solution

4 csc^{2} (\frac{x}{2}) - 6 csc (\frac{x}{2}) - 4 = 0

Solution

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Degrees

x = 6 0^{\circ} + 72 0^{\circ} n, x = 30 0^{\circ} + 72 0^{\circ} n

Solution steps

4 csc^{2} (\frac{x}{2}) - 6 csc (\frac{x}{2}) - 4 = 0

Solve by substitution

4 csc^{2} (\frac{x}{2}) - 6 csc (\frac{x}{2}) - 4 = 0

Let:

csc (\frac{x}{2}) = u

4 u^{2} - 6 u - 4 = 0

4 u^{2} - 6 u - 4 = 0 : u = 2, u = - \frac{1}{2}

4 u^{2} - 6 u - 4 = 0

Solve with the quadratic formula

4 u^{2} - 6 u - 4 = 0

Quadratic Equation Formula:

For

a = 4, b = - 6, c = - 4

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

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

(- 6)^{2} - 4 \cdot 4 (- 4) = 10

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 4 \cdot 4

Multiply the numbers:

4 \cdot 4 \cdot 4 = 64

= 6^{2} + 64

6^{2} = 36

= 36 + 64

Add the numbers:

36 + 64 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 6 ) \pm 10}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 10}{2 \cdot 4}, u_{2} = \frac{- ( - 6 ) - 10}{2 \cdot 4}

u = \frac{- ( - 6 ) + 10}{2 \cdot 4} : 2

\frac{- ( - 6 ) + 10}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{6 + 10}{2 \cdot 4}

Add the numbers:

6 + 10 = 16

= \frac{16}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= 2

u = \frac{- ( - 6 ) - 10}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( - 6 ) - 10}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{6 - 10}{2 \cdot 4}

Subtract the numbers:

6 - 10 = - 4

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = csc (\frac{x}{2})

csc (\frac{x}{2}) = 2, csc (\frac{x}{2}) = - \frac{1}{2}

csc (\frac{x}{2}) = 2, csc (\frac{x}{2}) = - \frac{1}{2}

csc (\frac{x}{2}) = 2 : x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

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

General solutions for

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

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

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

Solve

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

Solve

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

csc (\frac{x}{2}) = - \frac{1}{2} :

No Solution

csc (\frac{x}{2}) = - \frac{1}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4csc^2(x/2)-6csc(x/2)-4=0 ?
The general solution for 4csc^2(x/2)-6csc(x/2)-4=0 is x= pi/3+4pin,x=(5pi)/3+4pin