What is the general solution for 2sin^2(x)+3sin(x)=0 ?

The general solution for 2sin^2(x)+3sin(x)=0 is x=2pin,x=pi+2pin

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2sin^2(x)+3sin(x)=0

Solution

2 sin^{2} (x) + 3 sin (x) = 0

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (x) + 3 sin (x) = 0

Solve by substitution

2 sin^{2} (x) + 3 sin (x) = 0

Let:

sin (x) = u

2 u^{2} + 3 u = 0

2 u^{2} + 3 u = 0 : u = 0, u = - \frac{3}{2}

2 u^{2} + 3 u = 0

Solve with the quadratic formula

2 u^{2} + 3 u = 0

Quadratic Equation Formula:

For

a = 2, b = 3, c = 0

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

3^{2} - 4 \cdot 2 \cdot 0 = 3

3^{2} - 4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 3^{2} - 0

3^{2} - 0 = 3^{2}

= 3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

u_{1, 2} = \frac{- 3 \pm 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 3}{2 \cdot 2}, u_{2} = \frac{- 3 - 3}{2 \cdot 2}

u = \frac{- 3 + 3}{2 \cdot 2} : 0

\frac{- 3 + 3}{2 \cdot 2}

Add/Subtract the numbers:

- 3 + 3 = 0

= \frac{0}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

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

= 0

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

\frac{- 3 - 3}{2 \cdot 2}

Subtract the numbers:

- 3 - 3 = - 6

= \frac{- 6}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 6}{4}

Apply the fraction rule:

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

= - \frac{6}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = 0, u = - \frac{3}{2}

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - \frac{3}{2}

sin (x) = 0, sin (x) = - \frac{3}{2}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = - \frac{3}{2} :

No Solution

sin (x) = - \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

sin^2(x)=-sin(x)

sin^{2} (x) = - sin (x)

arccos(x)+2arcsin((sqrt(3))/2)=pi

arccos (x) + 2 arcsin (\frac{3}{2}) = π

3-3sin(θ)=2cos^2(θ)

3 - 3 sin (θ) = 2 cos^{2} (θ)

tan(a)=-sqrt(3)

tan (a) = - 3

cos^2(θ)=sin^2(θ)

cos^{2} (θ) = sin^{2} (θ)

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)+3sin(x)=0 ?
The general solution for 2sin^2(x)+3sin(x)=0 is x=2pin,x=pi+2pin