What is the general solution for 1+sin(x)=3sin(x) ?

The general solution for 1+sin(x)=3sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

1+sin(x)=3sin(x)

1 + sin (x) = 3 sin (x)

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

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

1 + sin (x) = 3 sin (x)

Solve by substitution

1 + sin (x) = 3 sin (x)

Let:

sin (x) = u

1 + u = 3 u

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

1 + u = 3 u

Move

1

to the right side

1 + u = 3 u

Subtract

1

from both sides

1 + u - 1 = 3 u - 1

Simplify

u = 3 u - 1

u = 3 u - 1

Move

3 u

to the left side

u = 3 u - 1

Subtract

3 u

from both sides

u - 3 u = 3 u - 1 - 3 u

Simplify

- 2 u = - 1

- 2 u = - 1

Divide both sides by

- 2

- 2 u = - 1

Divide both sides by

- 2

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

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = sin (x)

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

General solutions for

sin (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}

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

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

Combine all the solutions

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

3 cos (3 x) = 0

3 tan (x) = 2 cos (x)

6 tan^{3} (x) = 2 tan (x)

tan^{2} (θ) = tan (θ)

sin (2 θ) = - sin (θ)

What is the general solution for 1+sin(x)=3sin(x) ?
The general solution for 1+sin(x)=3sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin