What is the general solution for 3tan^3(x)-1=0 ?

The general solution for 3tan^3(x)-1=0 is x=0.60625…+pin

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3tan^3(x)-1=0

Solution

3 tan^{3} (x) - 1 = 0

Solution

x = 0.60625 \dots + πn

Degrees

x = 34.73594 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{3} (x) - 1 = 0

Solve by substitution

3 tan^{3} (x) - 1 = 0

Let:

tan (x) = u

3 u^{3} - 1 = 0

3 u^{3} - 1 = 0

Move

1

to the right side

3 u^{3} - 1 = 0

Add

1

to both sides

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

Simplify

3 u^{3} = 1

3 u^{3} = 1

Divide both sides by

3

3 u^{3} = 1

Divide both sides by

3

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

Simplify

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

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

For

x^{3} = f (a)

the solutions are

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

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

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

Rewrite

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

in standard complex form:

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

Apply radical rule:

Apply the fraction rule:

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

Cancel

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

Apply radical rule:

Multiply by the conjugate

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

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

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

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

Rewrite

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

in standard complex form:

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

Apply radical rule:

Apply the fraction rule:

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

Cancel

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

Apply radical rule:

Multiply by the conjugate

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

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

Substitute back

u = tan (x)

Apply trig inverse properties

General solutions for

tan (x) = a \Rightarrow x = arctan (a) + πn

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

x = 0.60625 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan^3(x)-1=0 ?
The general solution for 3tan^3(x)-1=0 is x=0.60625…+pin