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

The general solution for tan^3(x)=1 is x= pi/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan^3(x)=1

tan^{3} (x) = 1

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

tan^{3} (x) = 1

Solve by substitution

tan^{3} (x) = 1

Let:

tan (x) = u

u^{3} = 1

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

u^{3} = 1

For

x^{3} = f (a)

the solutions are

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

Simplify

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

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

Rewrite

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

in standard complex form:

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

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

Apply the fraction rule:

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

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

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

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

Simplify

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

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

Rewrite

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

in standard complex form:

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

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

Apply the fraction rule:

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

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

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

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

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

Substitute back

u = tan (x)

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

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

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

No Solution

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

No Solution

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

No Solution

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

No Solution

Combine all the solutions

x = \frac{π}{4} + πn

sin^{2} (x) - 9 cos (x) - 9 = 0

sin (a) = \frac{8}{17}

cot (t) = 1

4 sin (x) + 3 = 5

tan (θ) = \frac{3}{3}

What is the general solution for tan^3(x)=1 ?
The general solution for tan^3(x)=1 is x= pi/4+pin