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

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

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3sin(x)=sqrt(3)+sin(x)

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

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

3 u = 3 + u

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

3 u = 3 + u

Move

u

to the left side

3 u = 3 + u

Subtract

u

from both sides

3 u - u = 3 + u - u

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

\frac{2 u}{2} = \frac{3}{2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

Substitute back

u = sin (x)

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

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

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

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

General solutions for

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

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

Combine all the solutions

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

cos^{2} (x) = 2 + cos (x)

2 sin (θ) = 2 sin (θ) cos (θ)

cos (3 x) = sin (6 0^{\circ})

tan (x) + sec (x) = 3

2 cos^{2} (x) - 7 sin (x) - 5 = 0

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