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

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

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

Solution

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

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

Solve by substitution

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

Let:

sin (x) = u

u^{2} + 3 u = 0

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

u^{2} + 3 u = 0

Solve with the quadratic formula

u^{2} + 3 u = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = 0

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

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

3^{2} - 4 \cdot 1 \cdot 0 = 3

3^{2} - 4 \cdot 1 \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 1}

Separate the solutions

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

u = \frac{- 3 + 3}{2 \cdot 1} : 0

\frac{- 3 + 3}{2 \cdot 1}

Add/Subtract the numbers:

- 3 + 3 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

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

= 0

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

\frac{- 3 - 3}{2 \cdot 1}

Subtract the numbers:

- 3 - 3 = - 6

= \frac{- 6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = 0, u = - 3

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - 3

sin (x) = 0, sin (x) = - 3

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) = - 3 :

No Solution

sin (x) = - 3

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

sin(x)=0.62

sin (x) = 0.62

sin(x)=0.59

sin (x) = 0.59

sin(x)=0.74

sin (x) = 0.74

sin(x)=0.72

sin (x) = 0.72

sin(x)=0.28

sin (x) = 0.28

Frequently Asked Questions (FAQ)

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