What is the general solution for 3cot^2(x)-9=0 ?

The general solution for 3cot^2(x)-9=0 is x= pi/6+pin,x=(5pi)/6+pin

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3cot^2(x)-9=0

3 cot^{2} (x) - 9 = 0

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

3 cot^{2} (x) - 9 = 0

Solve by substitution

3 cot^{2} (x) - 9 = 0

Let:

cot (x) = u

3 u^{2} - 9 = 0

3 u^{2} - 9 = 0 : u = 3, u = - 3

3 u^{2} - 9 = 0

Move

9

to the right side

3 u^{2} - 9 = 0

Add

9

to both sides

3 u^{2} - 9 + 9 = 0 + 9

Simplify

3 u^{2} = 9

3 u^{2} = 9

Divide both sides by

3

3 u^{2} = 9

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{9}{3}

Simplify

u^{2} = 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

Substitute back

u = cot (x)

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

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

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

Combine all the solutions

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

\frac{2}{5} = cos (θ)

2 - 2 cos^{2} (x) = sin (x)

cos (x) = \frac{6}{7}

2 cos (x) + 1 = sin (x)

csc^{2} (x) = 2 csc (x)

What is the general solution for 3cot^2(x)-9=0 ?
The general solution for 3cot^2(x)-9=0 is x= pi/6+pin,x=(5pi)/6+pin