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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

6sin(θ)=2+2sin(θ)

6 sin (θ) = 2 + 2 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

6 sin (θ) = 2 + 2 sin (θ)

Solve by substitution

6 sin (θ) = 2 + 2 sin (θ)

Let:

sin (θ) = u

6 u = 2 + 2 u

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

6 u = 2 + 2 u

Move

2 u

to the left side

6 u = 2 + 2 u

Subtract

2 u

from both sides

6 u - 2 u = 2 + 2 u - 2 u

Simplify

4 u = 2

4 u = 2

Divide both sides by

4

4 u = 2

Divide both sides by

4

\frac{4 u}{4} = \frac{2}{4}

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

sin (x - \frac{π}{4}) - sin (x + \frac{π}{4}) = 0.25

sin^{2} (θ) + cos^{2} (θ) = 2

sec (θ) - 4.905 = 0

csc (\frac{x}{2}) = 2

5 tan (b) + 7 = 2 tan (b) + 4

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