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

The general solution for tan(x)+tan^3(x)=0 is x=pin

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tan(x)+tan^3(x)=0

Solution

tan (x) + tan^{3} (x) = 0

Solution

x = πn

Degrees

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

Solution steps

tan (x) + tan^{3} (x) = 0

Solve by substitution

tan (x) + tan^{3} (x) = 0

Let:

tan (x) = u

u + u^{3} = 0

u + u^{3} = 0 : u = 0, u = i, u = - i

u + u^{3} = 0

Factor

u + u^{3} : u (u^{2} + 1)

u + u^{3}

Apply exponent rule:

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

u^{3} = u^{2} u

= u^{2} u + u

Factor out common term

u

= u (u^{2} + 1)

u (u^{2} + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u^{2} + 1 = 0

Solve

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

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

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

The solutions are

u = 0, u = i, u = - i

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = i, tan (x) = - i

tan (x) = 0, tan (x) = i, tan (x) = - i

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

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 = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

tan (x) = i :

No Solution

tan (x) = i

No Solution

tan (x) = - i :

No Solution

tan (x) = - i

No Solution

Combine all the solutions

x = πn

Popular Examples

sin^3(x)-2sin(x)=4cos^2(x)-3

sin^{3} (x) - 2 sin (x) = 4 cos^{2} (x) - 3

cos^2(x)+6sin(x)-6=0

cos^{2} (x) + 6 sin (x) - 6 = 0

(1-sin(x))*cos^3(x)=0

(1 - sin (x)) \cdot cos^{3} (x) = 0

-tan(x)=3.465

- tan (x) = 3.465

3cos^2(x)-sin(x)+1=0

3 cos^{2} (x) - sin (x) + 1 = 0

Frequently Asked Questions (FAQ)

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