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

The general solution for 3cot^2(2x-(3pi)/2)=1 is x=(pin)/2+(11pi}{12},x=(pin)/2+\frac{13pi)/(12)

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

Solution

3 cot^{2} (2 x - \frac{3 π}{2}) = 1

Solution

x = \frac{πn}{2} + \frac{11 π}{12}, x = \frac{πn}{2} + \frac{13 π}{12}

Degrees

x = 16 5^{\circ} + 9 0^{\circ} n, x = 19 5^{\circ} + 9 0^{\circ} n

Solution steps

3 cot^{2} (2 x - \frac{3 π}{2}) = 1

Solve by substitution

3 cot^{2} (2 x - \frac{3 π}{2}) = 1

Let:

cot (2 x - \frac{3 π}{2}) = u

3 u^{2} = 1

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

3 u^{2} = 1

Divide both sides by

3

3 u^{2} = 1

Divide both sides by

3

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

Simplify

u^{2} = \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{3 π}{2})

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

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

cot (2 x - \frac{3 π}{2}) = \frac{1}{3} : x = \frac{πn}{2} + \frac{11 π}{12}

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

2 x - \frac{3 π}{2} = arccot (\frac{1}{3}) + πn : x = \frac{πn}{2} + \frac{11 π}{12}

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

Move

\frac{3 π}{2}

to the right side

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

Add

\frac{3 π}{2}

to both sides

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

Simplify

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

Simplify

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

2 x - \frac{3 π}{2} + \frac{3 π}{2}

Add similar elements:

- \frac{3 π}{2} + \frac{3 π}{2} = 0

= 2 x

Simplify

\frac{π}{3} + πn + \frac{3 π}{2} : πn + \frac{11 π}{6}

\frac{π}{3} + πn + \frac{3 π}{2}

Group like terms

= πn + \frac{π}{3} + \frac{3 π}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

For

\frac{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= \frac{π 2}{6} + \frac{9 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 9 π}{6}

Add similar elements:

2 π + 9 π = 11 π

= πn + \frac{11 π}{6}

2 x = πn + \frac{11 π}{6}

2 x = πn + \frac{11 π}{6}

2 x = πn + \frac{11 π}{6}

Divide both sides by

2

2 x = πn + \frac{11 π}{6}

Divide both sides by

2

\frac{2 x}{2} = \frac{πn}{2} + \frac{\frac{11 π}{6}}{2}

Simplify

\frac{2 x}{2} = \frac{πn}{2} + \frac{\frac{11 π}{6}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{πn}{2} + \frac{\frac{11 π}{6}}{2} : \frac{πn}{2} + \frac{11 π}{12}

\frac{πn}{2} + \frac{\frac{11 π}{6}}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

= \frac{πn}{2} + \frac{11 π}{12}

x = \frac{πn}{2} + \frac{11 π}{12}

x = \frac{πn}{2} + \frac{11 π}{12}

x = \frac{πn}{2} + \frac{11 π}{12}

x = \frac{πn}{2} + \frac{11 π}{12}

cot (2 x - \frac{3 π}{2}) = - \frac{1}{3} : x = \frac{πn}{2} + \frac{13 π}{12}

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

2 x - \frac{3 π}{2} = arccot (- \frac{1}{3}) + πn : x = \frac{πn}{2} + \frac{13 π}{12}

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

Move

\frac{3 π}{2}

to the right side

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

Add

\frac{3 π}{2}

to both sides

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

Simplify

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

Simplify

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

2 x - \frac{3 π}{2} + \frac{3 π}{2}

Add similar elements:

- \frac{3 π}{2} + \frac{3 π}{2} = 0

= 2 x

Simplify

\frac{2 π}{3} + πn + \frac{3 π}{2} : πn + \frac{13 π}{6}

\frac{2 π}{3} + πn + \frac{3 π}{2}

Group like terms

= πn + \frac{2 π}{3} + \frac{3 π}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

For

\frac{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= \frac{4 π}{6} + \frac{9 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π + 9 π}{6}

Add similar elements:

4 π + 9 π = 13 π

= πn + \frac{13 π}{6}

2 x = πn + \frac{13 π}{6}

2 x = πn + \frac{13 π}{6}

2 x = πn + \frac{13 π}{6}

Divide both sides by

2

2 x = πn + \frac{13 π}{6}

Divide both sides by

2

\frac{2 x}{2} = \frac{πn}{2} + \frac{\frac{13 π}{6}}{2}

Simplify

\frac{2 x}{2} = \frac{πn}{2} + \frac{\frac{13 π}{6}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{πn}{2} + \frac{\frac{13 π}{6}}{2} : \frac{πn}{2} + \frac{13 π}{12}

\frac{πn}{2} + \frac{\frac{13 π}{6}}{2}

\frac{\frac{13 π}{6}}{2} = \frac{13 π}{12}

\frac{\frac{13 π}{6}}{2}

Apply the fraction rule:

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

= \frac{13 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{13 π}{12}

= \frac{πn}{2} + \frac{13 π}{12}

x = \frac{πn}{2} + \frac{13 π}{12}

x = \frac{πn}{2} + \frac{13 π}{12}

x = \frac{πn}{2} + \frac{13 π}{12}

x = \frac{πn}{2} + \frac{13 π}{12}

Combine all the solutions

x = \frac{πn}{2} + \frac{11 π}{12}, x = \frac{πn}{2} + \frac{13 π}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(2x-(3pi)/2)=1 ?
The general solution for 3cot^2(2x-(3pi)/2)=1 is x=(pin)/2+(11pi}{12},x=(pin)/2+\frac{13pi)/(12)