What is the general solution for cot^3(x)+2cot^2(x)-3cot(x)-6=0 ?

The general solution for cot^3(x)+2cot^2(x)-3cot(x)-6=0 is x=2.67794…+pin,x=(5pi)/6+pin,x= pi/6+pin

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cot^3(x)+2cot^2(x)-3cot(x)-6=0

Solution

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Solution

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Degrees

x = 153.43494 \dots^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Solve by substitution

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Let:

cot (x) = u

u^{3} + 2 u^{2} - 3 u - 6 = 0

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

u^{3} + 2 u^{2} - 3 u - 6 = 0

Factor

u^{3} + 2 u^{2} - 3 u - 6 : (u + 2) (u + 3) (u - 3)

u^{3} + 2 u^{2} - 3 u - 6

= (u^{3} + 2 u^{2}) + (- 3 u - 6)

Factor out

- 3

from

- 3 u - 6 : - 3 (u + 2)

- 3 u - 6

Rewrite

6

3 \cdot 2

= - 3 u - 3 \cdot 2

Factor out common term

- 3

= - 3 (u + 2)

Factor out

u^{2}

from

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

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

Apply exponent rule:

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

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

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

Factor out common term

u^{2}

= u^{2} (u + 2)

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

Factor out common term

u + 2

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

Factor

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

u^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (u + 3) (u - 3)

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

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 + 3 = 0 : u = - 3

u + 3 = 0

Move

3

to the right side

u + 3 = 0

Subtract

3

from both sides

u + 3 - 3 = 0 - 3

Simplify

u = - 3

u = - 3

Solve

u - 3 = 0 : u = 3

u - 3 = 0

Move

3

to the right side

u - 3 = 0

Add

3

to both sides

u - 3 + 3 = 0 + 3

Simplify

u = 3

u = 3

The solutions are

u = - 2, u = - 3, u = 3

Substitute back

u = cot (x)

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2 : x = arccot (- 2) + πn

cot (x) = - 2

Apply trig inverse properties

cot (x) = - 2

General solutions for

cot (x) = - 2

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2) + πn

x = arccot (- 2) + πn

cot (x) = - 3 : x = \frac{5 π}{6} + πn

cot (x) = - 3

General solutions for

cot (x) = - 3

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

General solutions for

cot (x) = 3

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = arccot (- 2) + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Show solutions in decimal form

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cot^3(x)+2cot^2(x)-3cot(x)-6=0 ?
The general solution for cot^3(x)+2cot^2(x)-3cot(x)-6=0 is x=2.67794…+pin,x=(5pi)/6+pin,x= pi/6+pin