What is the general solution for 8sin(θ)+1=6sin(θ) ?

The general solution for 8sin(θ)+1=6sin(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin

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8sin(θ)+1=6sin(θ)

8 sin (θ) + 1 = 6 sin (θ)

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

Degrees

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

Solution steps

8 sin (θ) + 1 = 6 sin (θ)

Solve by substitution

8 sin (θ) + 1 = 6 sin (θ)

Let:

sin (θ) = u

8 u + 1 = 6 u

8 u + 1 = 6 u : u = - \frac{1}{2}

8 u + 1 = 6 u

Move

1

to the right side

8 u + 1 = 6 u

Subtract

1

from both sides

8 u + 1 - 1 = 6 u - 1

Simplify

8 u = 6 u - 1

8 u = 6 u - 1

Move

6 u

to the left side

8 u = 6 u - 1

Subtract

6 u

from both sides

8 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{7 π}{6} + 2 πn, θ = \frac{11 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{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

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

Combine all the solutions

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

- cot^{2} (x) = 1 + 2 cot (x)

6 sec (2 x) + 3 tan (2 x) - 9 = 0

3 cos^{2} (x) + 3 = 4, 0 \leq x < 2 π

tan (2 x) = 3

3 = tan (x)

What is the general solution for 8sin(θ)+1=6sin(θ) ?
The general solution for 8sin(θ)+1=6sin(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin