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

The general solution for cot^2(x)=(tan(x))/2 is x=0.89990…+pin

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cot^2(x)=(tan(x))/2

Solution

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

Solution

x = 0.89990 \dots + πn

Degrees

x = 51.56095 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

\frac{tan ( x )}{2}

from both sides

cot^{2} (x) - \frac{tan ( x )}{2} = 0

Simplify

cot^{2} (x) - \frac{tan ( x )}{2} : \frac{2 cot ^{2} ( x ) - tan ( x )}{2}

cot^{2} (x) - \frac{tan ( x )}{2}

Convert element to fraction:

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

= \frac{cot ^{2} ( x ) \cdot 2}{2} - \frac{tan ( x )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{cot ^{2} ( x ) \cdot 2 - tan ( x )}{2}

\frac{2 cot ^{2} ( x ) - tan ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cot^{2} (x) - tan (x) = 0

Rewrite using trig identities

- tan (x) + 2 cot^{2} (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

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

- \frac{1}{cot ( x )} + 2 cot^{2} (x) = 0

Solve by substitution

- \frac{1}{cot ( x )} + 2 cot^{2} (x) = 0

Let:

cot (x) = u

- \frac{1}{u} + 2 u^{2} = 0

- \frac{1}{u} + 2 u^{2} = 0 : u = 3 \frac{1}{2}, u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

- \frac{1}{u} + 2 u^{2} = 0

Multiply both sides by

u

- \frac{1}{u} + 2 u^{2} = 0

Multiply both sides by

u

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

Simplify

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

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

2 u^{2} u : 2 u^{3}

2 u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 1 + 2 u^{3} = 0

- 1 + 2 u^{3} = 0

- 1 + 2 u^{3} = 0

Solve

- 1 + 2 u^{3} = 0 : u = 3 \frac{1}{2}, u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

- 1 + 2 u^{3} = 0

Move

1

to the right side

- 1 + 2 u^{3} = 0

Add

1

to both sides

- 1 + 2 u^{3} + 1 = 0 + 1

Simplify

2 u^{3} = 1

2 u^{3} = 1

Divide both sides by

2

2 u^{3} = 1

Divide both sides by

2

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

Simplify

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

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

For

x^{3} = f (a)

the solutions are

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 \frac{1}{2}, u = 3 \frac{1}{2} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{2} \frac{- 1 - 3 i}{2}

Simplify

3 \frac{1}{2} \frac{- 1 + 3 i}{2} : - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}

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

Multiply fractions:

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

= \frac{( - 1 + 3 i ) 3 \frac{1}{2}}{2}

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

3 \frac{1}{2}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3 1}{3 2}

Apply rule

n 1 = 1

31 = 1

= \frac{1}{3 2}

= \frac{\frac{1}{3 2} ( - 1 + 3 i )}{2}

Multiply

(- 1 + 3 i) \frac{1}{3 2} : \frac{- 1 + 3 i}{3 2}

(- 1 + 3 i) \frac{1}{3 2}

Multiply fractions:

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

= \frac{1 \cdot ( - 1 + 3 i )}{3 2}

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

Multiply:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

Remove parentheses:

(- a) = - a

= - 1 + 3 i

= \frac{- 1 + 3 i}{3 2}

= \frac{\frac{- 1 + 3 i}{3 2}}{2}

Apply the fraction rule:

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

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

Rationalize

\frac{- 1 + 3 i}{2 3 2} : \frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{4}

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

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{( - 1 + 3 i ) \cdot 2 ^{\frac{2}{3}}}{3 2 \cdot 2 \cdot 2 ^{\frac{2}{3}}}

32 \cdot 2 \cdot 2^{\frac{2}{3}} = 4

32 \cdot 2 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{4}

= \frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{4}

Rewrite

\frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{4}

in standard complex form:

- \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

\frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{2 ^{2}}

Cancel

\frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{2 ^{2}} : \frac{- 1 + 3 i}{2 ^{\frac{4}{3}}}

\frac{2 ^{\frac{2}{3}} ( - 1 + 3 i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3 i}{2 ^{2 - \frac{2}{3}}}

Subtract the numbers:

2 - \frac{2}{3} = \frac{4}{3}

= \frac{- 1 + 3 i}{2 ^{\frac{4}{3}}}

= \frac{- 1 + 3 i}{2 ^{\frac{4}{3}}}

2^{\frac{4}{3}} = 232

2^{\frac{4}{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 232

= \frac{- 1 + 3 i}{2 3 2}

Apply the fraction rule:

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

\frac{- 1 + 3 i}{2 3 2} = - \frac{1}{2 3 2} + \frac{3 i}{2 3 2}

= - \frac{1}{2 3 2} + \frac{3 i}{2 3 2}

\frac{3}{2 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{4}

\frac{3}{2 3 2}

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4}

= - \frac{1}{2 3 2} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

- \frac{1}{2 3 2} = - \frac{2 ^{\frac{2}{3}}}{4}

- \frac{1}{2 3 2}

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

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

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{2}{3}}}{4}

= - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

= - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

Simplify

3 \frac{1}{2} \frac{- 1 - 3 i}{2} : - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

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

Multiply fractions:

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

= \frac{( - 1 - 3 i ) 3 \frac{1}{2}}{2}

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

3 \frac{1}{2}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3 1}{3 2}

Apply rule

n 1 = 1

31 = 1

= \frac{1}{3 2}

= \frac{\frac{1}{3 2} ( - 1 - 3 i )}{2}

Multiply

(- 1 - 3 i) \frac{1}{3 2} : \frac{- 1 - 3 i}{3 2}

(- 1 - 3 i) \frac{1}{3 2}

Multiply fractions:

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

= \frac{1 \cdot ( - 1 - 3 i )}{3 2}

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

Multiply:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

Remove parentheses:

(- a) = - a

= - 1 - 3 i

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

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

Apply the fraction rule:

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

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

Rationalize

\frac{- 1 - 3 i}{2 3 2} : \frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{4}

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

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{( - 1 - 3 i ) \cdot 2 ^{\frac{2}{3}}}{3 2 \cdot 2 \cdot 2 ^{\frac{2}{3}}}

32 \cdot 2 \cdot 2^{\frac{2}{3}} = 4

32 \cdot 2 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{4}

= \frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{4}

Rewrite

\frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{4}

in standard complex form:

- \frac{2 ^{\frac{2}{3}}}{4} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

\frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 ^{2}} : \frac{- 1 - 3 i}{2 ^{\frac{4}{3}}}

\frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 - 3 i}{2 ^{2 - \frac{2}{3}}}

Subtract the numbers:

2 - \frac{2}{3} = \frac{4}{3}

= \frac{- 1 - 3 i}{2 ^{\frac{4}{3}}}

= \frac{- 1 - 3 i}{2 ^{\frac{4}{3}}}

2^{\frac{4}{3}} = 232

2^{\frac{4}{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 232

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

Apply the fraction rule:

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

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

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

- \frac{3}{2 3 2} = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4}

- \frac{3}{2 3 2}

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4}

= - \frac{1}{2 3 2} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

- \frac{1}{2 3 2} = - \frac{2 ^{\frac{2}{3}}}{4}

- \frac{1}{2 3 2}

Multiply by the conjugate

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

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

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

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

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 ^{\frac{2}{3}}}{4}

= - \frac{2 ^{\frac{2}{3}}}{4} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

= - \frac{2 ^{\frac{2}{3}}}{4} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4} i

u = 3 \frac{1}{2}, u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

u = 3 \frac{1}{2}, u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- \frac{1}{u} + 2 u^{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3 \frac{1}{2}, u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

Substitute back

u = cot (x)

cot (x) = 3 \frac{1}{2}, cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

cot (x) = 3 \frac{1}{2}, cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}, cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (3 \frac{1}{2}) + πn

x = arccot (3 \frac{1}{2}) + πn

cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4} :

No Solution

cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{2 ^{\frac{2}{3}} 3}{4}

No Solution

cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4} :

No Solution

cot (x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{2 ^{\frac{2}{3}} 3}{4}

No Solution

Combine all the solutions

x = arccot (3 \frac{1}{2}) + πn

Show solutions in decimal form

x = 0.89990 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cot^2(x)=(tan(x))/2 ?
The general solution for cot^2(x)=(tan(x))/2 is x=0.89990…+pin