What is the general solution for 2-sin(θ)=3sin(θ) ?

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

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Popular >

2-sin(θ)=3sin(θ)

Solution

2 - sin (θ) = 3 sin (θ)

Solution

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

2 - sin (θ) = 3 sin (θ)

Solve by substitution

2 - sin (θ) = 3 sin (θ)

Let:

sin (θ) = u

2 - u = 3 u

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

2 - u = 3 u

Move

2

to the right side

2 - u = 3 u

Subtract

2

from both sides

2 - u - 2 = 3 u - 2

Simplify

- u = 3 u - 2

- u = 3 u - 2

Move

3 u

to the left side

- u = 3 u - 2

Subtract

3 u

from both sides

- u - 3 u = 3 u - 2 - 3 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

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

Simplify

\frac{- 4 u}{- 4} : u

\frac{- 4 u}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4 u}{4}

Divide the numbers:

\frac{4}{4} = 1

= u

Simplify

\frac{- 2}{- 4} : \frac{1}{2}

\frac{- 2}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{1}{2}

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

Popular Examples

-cos(x/2)+sin(x/2)=sqrt(2/5)

- cos (\frac{x}{2}) + sin (\frac{x}{2}) = \frac{2}{5}

(35)/(sin(90))=(27)/(sin(y))

\frac{35}{sin ( 9 0 ^{\circ} )} = \frac{27}{sin ( y )}

4cos^2(x)+2cos(x)-2sqrt(2)cos(x)=sqrt(2)

4 cos^{2} (x) + 2 cos (x) - 22 cos (x) = 2

sin(x+20)*csc(70)=1

sin (x + 2 0^{\circ}) \cdot csc (7 0^{\circ}) = 1

pi/2 =(arcsin(x))/c

\frac{π}{2} = \frac{arcsin ( x )}{c}

Frequently Asked Questions (FAQ)

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