What is the general solution for cot^4(x)-9=0 ?

The general solution for cot^4(x)-9=0 is x= pi/6+pin,x=(5pi)/6+pin

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Popular >

cot^4(x)-9=0

Solution

cot^{4} (x) - 9 = 0

Solution

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

Degrees

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

Solution steps

cot^{4} (x) - 9 = 0

Solve by substitution

cot^{4} (x) - 9 = 0

Let:

cot (x) = u

u^{4} - 9 = 0

u^{4} - 9 = 0 : u = 3, u = - 3, u = 3 i, u = - 3 i

u^{4} - 9 = 0

Move

9

to the right side

u^{4} - 9 = 0

Add

9

to both sides

u^{4} - 9 + 9 = 0 + 9

Simplify

u^{4} = 9

u^{4} = 9

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = 9

Solve

v^{2} = 9 : v = 9, v = - 9

v^{2} = 9

For

(g (x))^{2} = f (a)

the solutions are

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

v = 9, v = - 9

v = 9, v = - 9

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 9 : u = 3, u = - 3

u^{2} = 9

Simplify

9 : 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

Solve

u^{2} = - 9 : u = 3 i, u = - 3 i

u^{2} = - 9

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u^{2} = - 3

For

x^{2} = f (a)

the solutions are

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

u = - 3, u = - - 3

Simplify

- 3 : 3 i

- 3

Apply radical rule:

- a = - 1 a

- 3 = - 1 3

= - 1 3

Apply imaginary number rule:

- 1 = i

= 3 i

Simplify

- - 3 : - 3 i

- - 3

Simplify

- 3 : 3 i

- 3

Apply radical rule:

- a = - 1 a

- 3 = - 1 3

= - 1 3

Apply imaginary number rule:

- 1 = i

= 3 i

= - 3 i

u = 3 i, u = - 3 i

The solutions are

u = 3, u = - 3, u = 3 i, u = - 3 i

Substitute back

u = cot (x)

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

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

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

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 i :

No Solution

cot (x) = 3 i

No Solution

cot (x) = - 3 i :

No Solution

cot (x) = - 3 i

No Solution

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for cot^4(x)-9=0 ?
The general solution for cot^4(x)-9=0 is x= pi/6+pin,x=(5pi)/6+pin