What is the general solution for 9sin^2(x)+18sin(x)+9=0 ?

The general solution for 9sin^2(x)+18sin(x)+9=0 is x=(3pi)/2+2pin

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9sin^2(x)+18sin(x)+9=0

9 sin^{2} (x) + 18 sin (x) + 9 = 0

x = \frac{3 π}{2} + 2 πn

Degrees

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

Solution steps

9 sin^{2} (x) + 18 sin (x) + 9 = 0

Solve by substitution

9 sin^{2} (x) + 18 sin (x) + 9 = 0

Let:

sin (x) = u

9 u^{2} + 18 u + 9 = 0

9 u^{2} + 18 u + 9 = 0 : u = - 1

9 u^{2} + 18 u + 9 = 0

Solve with the quadratic formula

9 u^{2} + 18 u + 9 = 0

Quadratic Equation Formula:

For

a = 9, b = 18, c = 9

u_{1, 2} = \frac{- 18 \pm 1 8 ^{2} - 4 \cdot 9 \cdot 9}{2 \cdot 9}

u_{1, 2} = \frac{- 18 \pm 1 8 ^{2} - 4 \cdot 9 \cdot 9}{2 \cdot 9}

1 8^{2} - 4 \cdot 9 \cdot 9 = 0

1 8^{2} - 4 \cdot 9 \cdot 9

Multiply the numbers:

4 \cdot 9 \cdot 9 = 324

= 1 8^{2} - 324

1 8^{2} = 324

= 324 - 324

Subtract the numbers:

324 - 324 = 0

= 0

u_{1, 2} = \frac{- 18 \pm 0}{2 \cdot 9}

u = \frac{- 18}{2 \cdot 9}

\frac{- 18}{2 \cdot 9} = - 1

\frac{- 18}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 18}{18}

Apply the fraction rule:

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

= - \frac{18}{18}

Apply rule

\frac{a}{a} = 1

= - 1

u = - 1

The solution to the quadratic equation is:

u = - 1

Substitute back

u = sin (x)

sin (x) = - 1

sin (x) = - 1

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{3 π}{2} + 2 πn

- 12 sin^{2} (θ) + 7 = 4

4 cos (θ) + 5 = 0

cos (θ) = \frac{60}{78}

18 sin^{2} (x) = 9

x, sin (x) = cos (y)

What is the general solution for 9sin^2(x)+18sin(x)+9=0 ?
The general solution for 9sin^2(x)+18sin(x)+9=0 is x=(3pi)/2+2pin