What is the general solution for sin(3x)-2=-3sin(3x) ?

The general solution for sin(3x)-2=-3sin(3x) is x= pi/(18)+(2pin)/3 ,x=(5pi}{18}+\frac{2pin)/3

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sin(3x)-2=-3sin(3x)

Solution

sin (3 x) - 2 = - 3 sin (3 x)

Solution

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

Degrees

x = 1 0^{\circ} + 12 0^{\circ} n, x = 5 0^{\circ} + 12 0^{\circ} n

Solution steps

sin (3 x) - 2 = - 3 sin (3 x)

Solve by substitution

sin (3 x) - 2 = - 3 sin (3 x)

Let:

sin (3 x) = u

u - 2 = - 3 u

u - 2 = - 3 u : u = \frac{1}{2}

u - 2 = - 3 u

Move

2

to the right side

u - 2 = - 3 u

Add

2

to both sides

u - 2 + 2 = - 3 u + 2

Simplify

u = - 3 u + 2

u = - 3 u + 2

Move

3 u

to the left side

u = - 3 u + 2

Add

3 u

to both sides

u + 3 u = - 3 u + 2 + 3 u

Simplify

4 u = 2

4 u = 2

Divide both sides by

4

4 u = 2

Divide both sides by

4

\frac{4 u}{4} = \frac{2}{4}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = sin (3 x)

sin (3 x) = \frac{1}{2}

sin (3 x) = \frac{1}{2}

sin (3 x) = \frac{1}{2} : x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

sin (3 x) = \frac{1}{2}

General solutions for

sin (3 x) = \frac{1}{2}

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}

3 x = \frac{π}{6} + 2 πn, 3 x = \frac{5 π}{6} + 2 πn

3 x = \frac{π}{6} + 2 πn, 3 x = \frac{5 π}{6} + 2 πn

Solve

3 x = \frac{π}{6} + 2 πn : x = \frac{π}{18} + \frac{2 πn}{3}

3 x = \frac{π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3} : \frac{π}{18} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} = \frac{π}{18}

\frac{\frac{π}{6}}{3}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{π}{18}

= \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

Solve

3 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{18} + \frac{2 πn}{3}

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3} : \frac{5 π}{18} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} = \frac{5 π}{18}

\frac{\frac{5 π}{6}}{3}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{5 π}{18}

= \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x)-2=-3sin(3x) ?
The general solution for sin(3x)-2=-3sin(3x) is x= pi/(18)+(2pin)/3 ,x=(5pi}{18}+\frac{2pin)/3