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

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

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

5 sin (y) = 3 sin (y) + 3

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

Degrees

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

Solution steps

5 sin (y) = 3 sin (y) + 3

Solve by substitution

5 sin (y) = 3 sin (y) + 3

Let:

sin (y) = 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 (y)

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

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

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

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

General solutions for

sin (y) = \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}

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

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

Combine all the solutions

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

2 sin^{2} (θ) + sin^{2} (θ) = 0

cos (θ) = - 0.42, 0^{\circ} \leq θ < 36 0^{\circ}

sin (2 x) = \frac{1}{2}, 0 < x < 2 π

4 sin^{2} (θ) + 2 cos (θ) = 3

\frac{1715}{70} = tan (4 5^{\circ} + \frac{θ}{2})

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