What is the general solution for 3sin(x)-2=5sin(x)-1 ?

The general solution for 3sin(x)-2=5sin(x)-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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3sin(x)-2=5sin(x)-1

3 sin (x) - 2 = 5 sin (x) - 1

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (x) - 2 = 5 sin (x) - 1

Solve by substitution

3 sin (x) - 2 = 5 sin (x) - 1

Let:

sin (x) = u

3 u - 2 = 5 u - 1

3 u - 2 = 5 u - 1 : u = - \frac{1}{2}

3 u - 2 = 5 u - 1

Move

2

to the right side

3 u - 2 = 5 u - 1

Add

2

to both sides

3 u - 2 + 2 = 5 u - 1 + 2

Simplify

3 u = 5 u + 1

3 u = 5 u + 1

Move

5 u

to the left side

3 u = 5 u + 1

Subtract

5 u

from both sides

3 u - 5 u = 5 u + 1 - 5 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{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

1 + cos (θ) = 3 cos (θ)

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

tan (x) = \frac{1}{3}

cos (θ) + 1 = 2 sin^{2} (θ)

4 sin (θ) + 3 = 0

What is the general solution for 3sin(x)-2=5sin(x)-1 ?
The general solution for 3sin(x)-2=5sin(x)-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin