What is the general solution for sin(x)+sin^2(x)+sin^3(x)+sin^4(x)=0 ?

The general solution for sin(x)+sin^2(x)+sin^3(x)+sin^4(x)=0 is x=2pin,x=pi+2pin,x=(3pi)/2+2pin

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sin(x)+sin^2(x)+sin^3(x)+sin^4(x)=0

Solution

sin (x) + sin^{2} (x) + sin^{3} (x) + sin^{4} (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + sin^{2} (x) + sin^{3} (x) + sin^{4} (x) = 0

Solve by substitution

sin (x) + sin^{2} (x) + sin^{3} (x) + sin^{4} (x) = 0

Let:

sin (x) = u

u + u^{2} + u^{3} + u^{4} = 0

u + u^{2} + u^{3} + u^{4} = 0 : u = 0, u = - 1, u = i, u = - i

u + u^{2} + u^{3} + u^{4} = 0

Factor

u + u^{2} + u^{3} + u^{4} : u (u + 1) (u^{2} + 1)

u + u^{2} + u^{3} + u^{4}

Factor out common term

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

u^{4} + u^{3} + u^{2} + u

Apply exponent rule:

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

u^{2} = uu

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

Factor out common term

u

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

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

Factor

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

u^{3} + u^{2} + u + 1

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

Factor out

u^{2}

from

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

u^{3} + u^{2}

Apply exponent rule:

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

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

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

Factor out common term

u^{2}

= u^{2} (u + 1)

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

Factor out common term

u + 1

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

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 = - 1, u = i, u = - i

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - 1, sin (x) = i, sin (x) = - i

sin (x) = 0, sin (x) = - 1, sin (x) = i, sin (x) = - i

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = i :

No Solution

sin (x) = i

No Solution

sin (x) = - i :

No Solution

sin (x) = - i

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+sin^2(x)+sin^3(x)+sin^4(x)=0 ?
The general solution for sin(x)+sin^2(x)+sin^3(x)+sin^4(x)=0 is x=2pin,x=pi+2pin,x=(3pi)/2+2pin