What is the general solution for 6sin(θ)+2=sin(θ) ?

The general solution for 6sin(θ)+2=sin(θ) is θ=-0.41151…+2pin,θ=pi+0.41151…+2pin

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6sin(θ)+2=sin(θ)

6 sin (θ) + 2 = sin (θ)

θ = - 0.41151 \dots + 2 πn, θ = π + 0.41151 \dots + 2 πn

Degrees

θ = - 23.57817 \dots^{\circ} + 36 0^{\circ} n, θ = 203.57817 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 sin (θ) + 2 = sin (θ)

Solve by substitution

6 sin (θ) + 2 = sin (θ)

Let:

sin (θ) = u

6 u + 2 = u

6 u + 2 = u : u = - \frac{2}{5}

6 u + 2 = u

Move

2

to the right side

6 u + 2 = u

Subtract

2

from both sides

6 u + 2 - 2 = u - 2

Simplify

6 u = u - 2

6 u = u - 2

Move

u

to the left side

6 u = u - 2

Subtract

u

from both sides

6 u - u = u - 2 - u

Simplify

5 u = - 2

5 u = - 2

Divide both sides by

5

5 u = - 2

Divide both sides by

5

\frac{5 u}{5} = \frac{- 2}{5}

Simplify

u = - \frac{2}{5}

u = - \frac{2}{5}

Substitute back

u = sin (θ)

sin (θ) = - \frac{2}{5}

sin (θ) = - \frac{2}{5}

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

sin (θ) = - \frac{2}{5}

Apply trig inverse properties

sin (θ) = - \frac{2}{5}

General solutions for

sin (θ) = - \frac{2}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

θ = arcsin (- \frac{2}{5}) + 2 πn, θ = π + arcsin (\frac{2}{5}) + 2 πn

θ = arcsin (- \frac{2}{5}) + 2 πn, θ = π + arcsin (\frac{2}{5}) + 2 πn

Combine all the solutions

θ = arcsin (- \frac{2}{5}) + 2 πn, θ = π + arcsin (\frac{2}{5}) + 2 πn

Show solutions in decimal form

θ = - 0.41151 \dots + 2 πn, θ = π + 0.41151 \dots + 2 πn

cos (θ) = - \frac{19}{17}

5 = 7 - cos (θ) + 5 cos (2 θ)

tan (2 x) = 0.32

sin (x) = \frac{- 3}{2}, 27 0^{\circ} < x < 36 0^{\circ}

x, R = \frac{v ^{2} sin ( 2 x )}{g}

What is the general solution for 6sin(θ)+2=sin(θ) ?
The general solution for 6sin(θ)+2=sin(θ) is θ=-0.41151…+2pin,θ=pi+0.41151…+2pin