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

The general solution for tan^3(x)=2 is x=0.89990…+pin

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tan^3(x)=2

Solution

tan^{3} (x) = 2

Solution

x = 0.89990 \dots + πn

Degrees

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

Solution steps

tan^{3} (x) = 2

Solve by substitution

tan^{3} (x) = 2

Let:

tan (x) = u

u^{3} = 2

u^{3} = 2

For

x^{3} = f (a)

the solutions are

Simplify

Multiply fractions:

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

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Rationalize

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Rewrite

in standard complex form:

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply the fraction rule:

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

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

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

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Simplify

Multiply fractions:

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

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Rationalize

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Rewrite

in standard complex form:

Apply radical rule:

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply the fraction rule:

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

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

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

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

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

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

\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{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

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 = 0.89990 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^3(x)=2 ?
The general solution for tan^3(x)=2 is x=0.89990…+pin