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

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

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

Solution

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

Solution

x = \frac{π}{2} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cot (x) = u

u^{2} = - 3 u

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

u^{2} = - 3 u

Move

3 u

to the left side

u^{2} = - 3 u

Add

3 u

to both sides

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

Simplify

u^{2} + 3 u = 0

u^{2} + 3 u = 0

Solve with the quadratic formula

u^{2} + 3 u = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = 0

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(3)^{2} - 4 \cdot 1 \cdot 0 = 3

(3)^{2} - 4 \cdot 1 \cdot 0

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 3 - 0

Subtract the numbers:

3 - 0 = 3

= 3

u_{1, 2} = \frac{- 3 \pm 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 3}{2 \cdot 1}, u_{2} = \frac{- 3 - 3}{2 \cdot 1}

u = \frac{- 3 + 3}{2 \cdot 1} : 0

\frac{- 3 + 3}{2 \cdot 1}

Add similar elements:

- 3 + 3 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 3 - 3}{2 \cdot 1} : - 3

\frac{- 3 - 3}{2 \cdot 1}

Add similar elements:

- 3 - 3 = - 23

= \frac{- 2 3}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2 3}{2}

Apply the fraction rule:

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

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

The solutions to the quadratic equation are:

u = 0, u = - 3

Substitute back

u = cot (x)

cot (x) = 0, cot (x) = - 3

cot (x) = 0, cot (x) = - 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) = - 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}{2} + πn, x = \frac{5 π}{6} + πn

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Frequently Asked Questions (FAQ)

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