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

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

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Popular >

sin^3(x)=3sin(x)

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

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

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u^{3} = 3 u

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

u^{3} = 3 u

Move

3 u

to the left side

u^{3} = 3 u

Subtract

3 u

from both sides

u^{3} - 3 u = 3 u - 3 u

Simplify

u^{3} - 3 u = 0

u^{3} - 3 u = 0

Factor

u^{3} - 3 u : u (u + 3) (u - 3)

u^{3} - 3 u

Factor out common term

u : u (u^{2} - 3)

u^{3} - 3 u

Apply exponent rule:

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

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

= u^{2} u - 3 u

Factor out common term

u

= u (u^{2} - 3)

= u (u^{2} - 3)

Factor

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

u^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= u^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (u + 3) (u - 3)

= u (u + 3) (u - 3)

u (u + 3) (u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 3 = 0 or u - 3 = 0

Solve

u + 3 = 0 : u = - 3

u + 3 = 0

Move

3

to the right side

u + 3 = 0

Subtract

3

from both sides

u + 3 - 3 = 0 - 3

Simplify

u = - 3

u = - 3

Solve

u - 3 = 0 : u = 3

u - 3 = 0

Move

3

to the right side

u - 3 = 0

Add

3

to both sides

u - 3 + 3 = 0 + 3

Simplify

u = 3

u = 3

The solutions are

u = 0, u = - 3, u = 3

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - 3, sin (x) = 3

sin (x) = 0, sin (x) = - 3, sin (x) = 3

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) = - 3 :

No Solution

sin (x) = - 3

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = 3 :

No Solution

sin (x) = 3

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

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