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

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

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cot^2(x/2)=3

Solution

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

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cot (\frac{x}{2}) = u

u^{2} = 3

u^{2} = 3 : u = 3, u = - 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 (\frac{x}{2})

cot (\frac{x}{2}) = 3, cot (\frac{x}{2}) = - 3

cot (\frac{x}{2}) = 3, cot (\frac{x}{2}) = - 3

cot (\frac{x}{2}) = 3 : x = \frac{π}{3} + 2 πn

cot (\frac{x}{2}) = 3

General solutions for

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

\frac{x}{2} = \frac{π}{6} + πn

\frac{x}{2} = \frac{π}{6} + πn

Solve

\frac{x}{2} = \frac{π}{6} + πn : x = \frac{π}{3} + 2 πn

\frac{x}{2} = \frac{π}{6} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{6} + 2 πn : \frac{π}{3} + 2 πn

2 \cdot \frac{π}{6} + 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

cot (\frac{x}{2}) = - 3 : x = \frac{5 π}{3} + 2 πn

cot (\frac{x}{2}) = - 3

General solutions for

cot (\frac{x}{2}) = - 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

\frac{x}{2} = \frac{5 π}{6} + πn

\frac{x}{2} = \frac{5 π}{6} + πn

Solve

\frac{x}{2} = \frac{5 π}{6} + πn : x = \frac{5 π}{3} + 2 πn

\frac{x}{2} = \frac{5 π}{6} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{6} + 2 πn : \frac{5 π}{3} + 2 πn

2 \cdot \frac{5 π}{6} + 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

= \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

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