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

The general solution for 2sin(x)+(sqrt(3))/2 =sin(x) is x=(4pi)/3+2pin,x=(5pi)/3+2pin

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2sin(x)+(sqrt(3))/2 =sin(x)

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

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Move

\frac{3}{2}

to the right side

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

Subtract

\frac{3}{2}

from both sides

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

Simplify

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

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

Move

u

to the left side

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

Subtract

u

from both sides

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

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{4 π}{3} + 2 πn, x = \frac{5 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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

3 tan (2 x) = 3

12 cos^{2} (x) = 3

2 cos (x) - 2 sin (x) = 1, 0 \leq x < 2 π

sin (x) = 3 cos (x)

2 sin^{2} (x) - 3 cos (x) = 3

What is the general solution for 2sin(x)+(sqrt(3))/2 =sin(x) ?
The general solution for 2sin(x)+(sqrt(3))/2 =sin(x) is x=(4pi)/3+2pin,x=(5pi)/3+2pin