What is the general solution for cot^2(2x-pi/3)= 1/3 ?

The general solution for cot^2(2x-pi/3)= 1/3 is x= pi/3+(pin)/2 ,x=(pin)/2+pi/2

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Popular Trigonometry >

cot^2(2x-pi/3)= 1/3

Solution

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Solution

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

Degrees

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

Solution steps

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Solve by substitution

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Let:

cot (2 x - \frac{π}{3}) = u

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

u^{2} = \frac{1}{3} : u = \frac{1}{3}, u = - \frac{1}{3}

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = cot (2 x - \frac{π}{3})

cot (2 x - \frac{π}{3}) = \frac{1}{3}, cot (2 x - \frac{π}{3}) = - \frac{1}{3}

cot (2 x - \frac{π}{3}) = \frac{1}{3}, cot (2 x - \frac{π}{3}) = - \frac{1}{3}

cot (2 x - \frac{π}{3}) = \frac{1}{3} : x = \frac{π}{3} + \frac{πn}{2}

cot (2 x - \frac{π}{3}) = \frac{1}{3}

Apply trig inverse properties

cot (2 x - \frac{π}{3}) = \frac{1}{3}

General solutions for

cot (2 x - \frac{π}{3}) = \frac{1}{3}

cot (x) = a \Rightarrow x = arccot (a) + πn

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

Solve

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn : x = \frac{π}{3} + \frac{πn}{2}

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

Simplify

arccot (\frac{1}{3}) + πn : \frac{π}{3} + πn

arccot (\frac{1}{3}) + πn

Use the following trivial identity:

arccot (\frac{1}{3}) = \frac{π}{3}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{π}{3} + πn

2 x - \frac{π}{3} = \frac{π}{3} + πn

Move

\frac{π}{3}

to the right side

2 x - \frac{π}{3} = \frac{π}{3} + πn

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

2 x - \frac{π}{3} + \frac{π}{3} : 2 x

2 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 x

Simplify

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

\frac{π}{3} + πn + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{π}{3} + πn

Combine the fractions

\frac{π}{3} + \frac{π}{3} : \frac{2 π}{3}

Apply rule

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

= \frac{π + π}{3}

Add similar elements:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

Divide both sides by

2

2 x = \frac{2 π}{3} + πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{\frac{2 π}{3}}{2} + \frac{πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

cot (2 x - \frac{π}{3}) = - \frac{1}{3} : x = \frac{πn}{2} + \frac{π}{2}

cot (2 x - \frac{π}{3}) = - \frac{1}{3}

Apply trig inverse properties

cot (2 x - \frac{π}{3}) = - \frac{1}{3}

General solutions for

cot (2 x - \frac{π}{3}) = - \frac{1}{3}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

Solve

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn : x = \frac{πn}{2} + \frac{π}{2}

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

Simplify

arccot (- \frac{1}{3}) + πn : \frac{2 π}{3} + πn

arccot (- \frac{1}{3}) + πn

Use the following trivial identity:

arccot (- \frac{1}{3}) = \frac{2 π}{3}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{2 π}{3} + πn

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

2 x - \frac{π}{3} + \frac{π}{3} : 2 x

2 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 x

Simplify

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

\frac{2 π}{3} + πn + \frac{π}{3}

Group like terms

= πn + \frac{π}{3} + \frac{2 π}{3}

Combine the fractions

\frac{π}{3} + \frac{2 π}{3} : π

Apply rule

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

= \frac{π + 2 π}{3}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= π

= πn + π

2 x = πn + π

2 x = πn + π

2 x = πn + π

Divide both sides by

2

2 x = πn + π

Divide both sides by

2

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

Simplify

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cot^2(2x-pi/3)= 1/3 ?
The general solution for cot^2(2x-pi/3)= 1/3 is x= pi/3+(pin)/2 ,x=(pin)/2+pi/2