What is the general solution for 5sin(x)=3sin(x)+sqrt(3) ?

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

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5sin(x)=3sin(x)+sqrt(3)

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

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

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

Solve by substitution

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

Let:

sin (x) = u

5 u = 3 u + 3

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

5 u = 3 u + 3

Move

3 u

to the left side

5 u = 3 u + 3

Subtract

3 u

from both sides

5 u - 3 u = 3 u + 3 - 3 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

9 cot^{2} (θ) - 16 = 0

cos (θ) = - \frac{3}{4}

sin^{2} (x) - 3 sin (x) + 2 = 0

sin (θ) = \frac{1}{5}

sin (4 x) = sin (2 x)

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