What is the general solution for 6sin^2(x)+9sin(x)+3=0 ?

The general solution for 6sin^2(x)+9sin(x)+3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=(3pi)/2+2pin

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Popular >

6sin^2(x)+9sin(x)+3=0

Solution

6 sin^{2} (x) + 9 sin (x) + 3 = 0

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

6 sin^{2} (x) + 9 sin (x) + 3 = 0

Solve by substitution

6 sin^{2} (x) + 9 sin (x) + 3 = 0

Let:

sin (x) = u

6 u^{2} + 9 u + 3 = 0

6 u^{2} + 9 u + 3 = 0 : u = - \frac{1}{2}, u = - 1

6 u^{2} + 9 u + 3 = 0

Solve with the quadratic formula

6 u^{2} + 9 u + 3 = 0

Quadratic Equation Formula:

For

a = 6, b = 9, c = 3

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

9^{2} - 4 \cdot 6 \cdot 3 = 3

9^{2} - 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

Subtract the numbers:

81 - 72 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 9 \pm 3}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- 9 + 3}{2 \cdot 6}, u_{2} = \frac{- 9 - 3}{2 \cdot 6}

u = \frac{- 9 + 3}{2 \cdot 6} : - \frac{1}{2}

\frac{- 9 + 3}{2 \cdot 6}

Add/Subtract the numbers:

- 9 + 3 = - 6

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

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 6}{12}

Apply the fraction rule:

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

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

u = \frac{- 9 - 3}{2 \cdot 6} : - 1

\frac{- 9 - 3}{2 \cdot 6}

Subtract the numbers:

- 9 - 3 = - 12

= \frac{- 12}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 12}{12}

Apply the fraction rule:

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

= - \frac{12}{12}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

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

Substitute back

u = sin (x)

sin (x) = - \frac{1}{2}, sin (x) = - 1

sin (x) = - \frac{1}{2}, sin (x) = - 1

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \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}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

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 = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin^2(x)+9sin(x)+3=0 ?
The general solution for 6sin^2(x)+9sin(x)+3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=(3pi)/2+2pin