What is the general solution for tan^5(x)+tan^3(x)=0 ?

The general solution for tan^5(x)+tan^3(x)=0 is x=pin

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tan^5(x)+tan^3(x)=0

Solution

tan^{5} (x) + tan^{3} (x) = 0

Solution

x = πn

Degrees

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

Solution steps

tan^{5} (x) + tan^{3} (x) = 0

Solve by substitution

tan^{5} (x) + tan^{3} (x) = 0

Let:

tan (x) = u

u^{5} + u^{3} = 0

u^{5} + u^{3} = 0 : u = 0, u = i, u = - i

u^{5} + u^{3} = 0

Factor

u^{5} + u^{3} : u^{3} (u^{2} + 1)

u^{5} + u^{3}

Apply exponent rule:

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

u^{5} = u^{2} u^{3}

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

Factor out common term

u^{3}

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

u^{3} (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

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

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

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

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

1/(cos(x))=1.98

\frac{1}{cos ( x )} = 1.98

2sin^2(3θ)-1=0

2 sin^{2} (3 θ) - 1 = 0

4sin(4θ-pi/3)+3=5

4 sin (4 θ - \frac{π}{3}) + 3 = 5

Frequently Asked Questions (FAQ)

What is the general solution for tan^5(x)+tan^3(x)=0 ?
The general solution for tan^5(x)+tan^3(x)=0 is x=pin