What is the general solution for 3cot^2(x)=sqrt(3)cot(x) ?

The general solution for 3cot^2(x)=sqrt(3)cot(x) is x= pi/2+pin,x= pi/3+pin

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3cot^2(x)=sqrt(3)cot(x)

Solution

3 cot^{2} (x) = 3 cot (x)

Solution

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

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

3 cot^{2} (x) = 3 cot (x)

Solve by substitution

3 cot^{2} (x) = 3 cot (x)

Let:

cot (x) = u

3 u^{2} = 3 u

3 u^{2} = 3 u : u = 0, u = \frac{3}{3}

3 u^{2} = 3 u

Move

3 u

to the left side

3 u^{2} = 3 u

Subtract

3 u

from both sides

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

Simplify

3 u^{2} - 3 u = 0

3 u^{2} - 3 u = 0

Factor

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

3 u^{2} - 3 u

Apply exponent rule:

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

u^{2} = uu

= 3 uu - 3 u

Factor out common term

u

= u (3 u - 1 \cdot 3)

Multiply the numbers:

1 \cdot 3 = 3

= u (3 u - 3)

u (3 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 3 u - 3 = 0

Solve

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

3 u - 3 = 0

Move

3

to the right side

3 u - 3 = 0

Add

3

to both sides

3 u - 3 + 3 = 0 + 3

Simplify

3 u = 3

3 u = 3

Divide both sides by

3

3 u = 3

Divide both sides by

3

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

Simplify

u = \frac{3}{3}

u = \frac{3}{3}

The solutions to the quadratic equation are:

u = 0, u = \frac{3}{3}

Substitute back

u = cot (x)

cot (x) = 0, cot (x) = \frac{3}{3}

cot (x) = 0, cot (x) = \frac{3}{3}

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

General solutions for

cot (x) = 0

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}{2} + πn

x = \frac{π}{2} + πn

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

cot (x) = \frac{3}{3}

General solutions for

cot (x) = \frac{3}{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}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

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

Popular Examples

sin(7x)=(sqrt(3))/2

sin (7 x) = \frac{3}{2}

-1.9sin(t)=0.4cos(2t)

- 1.9 sin (t) = 0.4 cos (2 t)

sin(x/2)=(sqrt(2))/2

sin (\frac{x}{2}) = \frac{2}{2}

sin(2t)=1

sin (2 t) = 1

sin^4(x)=sin^2(x)

sin^{4} (x) = sin^{2} (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(x)=sqrt(3)cot(x) ?
The general solution for 3cot^2(x)=sqrt(3)cot(x) is x= pi/2+pin,x= pi/3+pin