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Popular Trigonometry >

csc^3(x)+csc^2(x)= 4/3 csc(x)+4/3

Solution

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

Solution

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

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

Solve by substitution

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

Let:

csc (x) = u

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3} : u = - 1, u = - \frac{2 3}{3}, u = \frac{2 3}{3}

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

Multiply both sides by

3

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

Multiply both sides by

3

u^{3} \cdot 3 + u^{2} \cdot 3 = \frac{4}{3} u \cdot 3 + \frac{4}{3} \cdot 3

Simplify

3 u^{3} + 3 u^{2} = 4 u + 4

3 u^{3} + 3 u^{2} = 4 u + 4

Move

4

to the left side

3 u^{3} + 3 u^{2} = 4 u + 4

Subtract

4

from both sides

3 u^{3} + 3 u^{2} - 4 = 4 u + 4 - 4

Simplify

3 u^{3} + 3 u^{2} - 4 = 4 u

3 u^{3} + 3 u^{2} - 4 = 4 u

Move

4 u

to the left side

3 u^{3} + 3 u^{2} - 4 = 4 u

Subtract

4 u

from both sides

3 u^{3} + 3 u^{2} - 4 - 4 u = 4 u - 4 u

Simplify

3 u^{3} + 3 u^{2} - 4 - 4 u = 0

3 u^{3} + 3 u^{2} - 4 - 4 u = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

3 u^{3} + 3 u^{2} - 4 u - 4 = 0

Factor

3 u^{3} + 3 u^{2} - 4 u - 4 : (u + 1) (3 u + 2) (3 u - 2)

3 u^{3} + 3 u^{2} - 4 u - 4

= (3 u^{3} + 3 u^{2}) + (- 4 u - 4)

Factor out

- 4

from

- 4 u - 4 : - 4 (u + 1)

- 4 u - 4

Factor out common term

- 4

= - 4 (u + 1)

Factor out

3 u^{2}

from

3 u^{3} + 3 u^{2} : 3 u^{2} (u + 1)

3 u^{3} + 3 u^{2}

Apply exponent rule:

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

u^{3} = u u^{2}

= 3 u u^{2} + 3 u^{2}

Factor out common term

3 u^{2}

= 3 u^{2} (u + 1)

= - 4 (u + 1) + 3 u^{2} (u + 1)

Factor out common term

u + 1

= (u + 1) (3 u^{2} - 4)

Factor

3 u^{2} - 4 : (3 u + 2) (3 u - 2)

3 u^{2} - 4

Rewrite

3 u^{2} - 4

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

3 u^{2} - 4

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} u^{2} - 4

Rewrite

4

2^{2}

= (3)^{2} u^{2} - 2^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} u^{2} = (3 u)^{2}

= (3 u)^{2} - 2^{2}

= (3 u)^{2} - 2^{2}

Apply Difference of Two Squares Formula:

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

(3 u)^{2} - 2^{2} = (3 u + 2) (3 u - 2)

= (3 u + 2) (3 u - 2)

= (u + 1) (3 u + 2) (3 u - 2)

(u + 1) (3 u + 2) (3 u - 2) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u + 1 = 0 or 3 u + 2 = 0 or 3 u - 2 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

3 u + 2 = 0 : u = - \frac{2 3}{3}

3 u + 2 = 0

Move

2

to the right side

3 u + 2 = 0

Subtract

2

from both sides

3 u + 2 - 2 = 0 - 2

Simplify

3 u = - 2

3 u = - 2

Divide both sides by

3

3 u = - 2

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 2}{3} : - \frac{2 3}{3}

\frac{- 2}{3}

Apply the fraction rule:

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

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

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

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

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

Solve

3 u - 2 = 0 : u = \frac{2 3}{3}

3 u - 2 = 0

Move

2

to the right side

3 u - 2 = 0

Add

2

to both sides

3 u - 2 + 2 = 0 + 2

Simplify

3 u = 2

3 u = 2

Divide both sides by

3

3 u = 2

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

u = \frac{2 3}{3}

u = \frac{2 3}{3}

u = \frac{2 3}{3}

The solutions are

u = - 1, u = - \frac{2 3}{3}, u = \frac{2 3}{3}

Substitute back

u = csc (x)

csc (x) = - 1, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = - 1, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

General solutions for

csc (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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

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

General solutions for

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

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

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

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

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

General solutions for

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

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}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

Combine all the solutions

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

Graph

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