What is the general solution for sqrt(2)sin^2(θ)+sin(θ)=0 ?

The general solution for sqrt(2)sin^2(θ)+sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=(5pi)/4+2pin,θ=(7pi)/4+2pin

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sqrt(2)sin^2(θ)+sin(θ)=0

Solution

2 sin^{2} (θ) + sin (θ) = 0

Solution

θ = 2 πn, θ = π + 2 πn, θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (θ) + sin (θ) = 0

Solve by substitution

2 sin^{2} (θ) + sin (θ) = 0

Let:

sin (θ) = u

2 u^{2} + u = 0

2 u^{2} + u = 0 : u = 0, u = - \frac{2}{2}

2 u^{2} + u = 0

Solve with the quadratic formula

2 u^{2} + u = 0

Quadratic Equation Formula:

For

a = 2, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 \cdot 0}{2 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 \cdot 0}{2 2}

1^{2} - 42 \cdot 0 = 1

1^{2} - 42 \cdot 0

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 0 \cdot 2

Apply rule

0 \cdot a = 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 2}

Separate the solutions

u_{1} = \frac{- 1 + 1}{2 2}, u_{2} = \frac{- 1 - 1}{2 2}

u = \frac{- 1 + 1}{2 2} : 0

\frac{- 1 + 1}{2 2}

Add/Subtract the numbers:

- 1 + 1 = 0

= \frac{0}{2 2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

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

\frac{- 1 - 1}{2 2}

Subtract the numbers:

- 1 - 1 = - 2

= \frac{- 2}{2 2}

Apply the fraction rule:

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

= - \frac{2}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

The solutions to the quadratic equation are:

u = 0, u = - \frac{2}{2}

Substitute back

u = sin (θ)

sin (θ) = 0, sin (θ) = - \frac{2}{2}

sin (θ) = 0, sin (θ) = - \frac{2}{2}

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

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 = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

sin (θ) = - \frac{2}{2} : θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sin (θ) = - \frac{2}{2}

General solutions for

sin (θ) = - \frac{2}{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{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin^2(θ)+sin(θ)=0 ?
The general solution for sqrt(2)sin^2(θ)+sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=(5pi)/4+2pin,θ=(7pi)/4+2pin