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Popular >

csc^5(x)-4csc(x)=0

Solution

csc^{5} (x) - 4 csc (x) = 0

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

csc^{5} (x) - 4 csc (x) = 0

Solve by substitution

csc^{5} (x) - 4 csc (x) = 0

Let:

csc (x) = u

u^{5} - 4 u = 0

u^{5} - 4 u = 0 : u = 0, u = 2 i, u = - 2 i, u = - 2, u = 2

u^{5} - 4 u = 0

Factor

u^{5} - 4 u : u (u^{2} + 2) (u + 2) (u - 2)

u^{5} - 4 u

Factor out common term

u : u (u^{4} - 4)

u^{5} - 4 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{5} = u^{4} u

= u^{4} u - 4 u

Factor out common term

u

= u (u^{4} - 4)

= u (u^{4} - 4)

Factor

u^{4} - 4 : (u^{2} + 2) (u + 2) (u - 2)

u^{4} - 4

Rewrite

u^{4} - 4

(u^{2})^{2} - 2^{2}

u^{4} - 4

Rewrite

4

2^{2}

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

Apply exponent rule:

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

u^{4} = (u^{2})^{2}

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Factor

u^{2} - 2 : (u + 2) (u - 2)

u^{2} - 2

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (u + 2) (u - 2)

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

Move

2

to the right side

u^{2} + 2 = 0

Subtract

2

from both sides

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

Simplify

u^{2} = - 2

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Solve

u + 2 = 0 : u = - 2

u + 2 = 0

Move

2

to the right side

u + 2 = 0

Subtract

2

from both sides

u + 2 - 2 = 0 - 2

Simplify

u = - 2

u = - 2

Solve

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

The solutions are

u = 0, u = 2 i, u = - 2 i, u = - 2, u = 2

Substitute back

u = csc (x)

csc (x) = 0, csc (x) = 2 i, csc (x) = - 2 i, csc (x) = - 2, csc (x) = 2

csc (x) = 0, csc (x) = 2 i, csc (x) = - 2 i, csc (x) = - 2, csc (x) = 2

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

csc (x) = 2 i :

No Solution

csc (x) = 2 i

No Solution

csc (x) = - 2 i :

No Solution

csc (x) = - 2 i

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

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

Combine all the solutions

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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