What is the general solution for 3-4sin(θ)=4-6sin(θ) ?

The general solution for 3-4sin(θ)=4-6sin(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin

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3-4sin(θ)=4-6sin(θ)

3 - 4 sin (θ) = 4 - 6 sin (θ)

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Degrees

θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

3 - 4 sin (θ) = 4 - 6 sin (θ)

Solve by substitution

3 - 4 sin (θ) = 4 - 6 sin (θ)

Let:

sin (θ) = u

3 - 4 u = 4 - 6 u

3 - 4 u = 4 - 6 u : u = \frac{1}{2}

3 - 4 u = 4 - 6 u

Move

3

to the right side

3 - 4 u = 4 - 6 u

Subtract

3

from both sides

3 - 4 u - 3 = 4 - 6 u - 3

Simplify

- 4 u = - 6 u + 1

- 4 u = - 6 u + 1

Move

6 u

to the left side

- 4 u = - 6 u + 1

Add

6 u

to both sides

- 4 u + 6 u = - 6 u + 1 + 6 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 (θ)

sin (θ) = \frac{1}{2}

sin (θ) = \frac{1}{2}

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = \frac{1}{2}

General solutions for

sin (θ) = \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}

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

\frac{cos ( x )}{sin ( x )} = - 0.00172

x, r^{3} = sin (x)

\frac{( 7 cos ( 2 x ) + 7 )}{2} = \frac{π}{2}

tan (x) = - 1.036

tan (2 x) = \frac{24}{7}

What is the general solution for 3-4sin(θ)=4-6sin(θ) ?
The general solution for 3-4sin(θ)=4-6sin(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin