What is the general solution for 7sin^2(x)+14sin(x)+7=0 ?

The general solution for 7sin^2(x)+14sin(x)+7=0 is x=(3pi)/2+2pin

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7sin^2(x)+14sin(x)+7=0

7 sin^{2} (x) + 14 sin (x) + 7 = 0

x = \frac{3 π}{2} + 2 πn

Degrees

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

Solution steps

7 sin^{2} (x) + 14 sin (x) + 7 = 0

Solve by substitution

7 sin^{2} (x) + 14 sin (x) + 7 = 0

Let:

sin (x) = u

7 u^{2} + 14 u + 7 = 0

7 u^{2} + 14 u + 7 = 0 : u = - 1

7 u^{2} + 14 u + 7 = 0

Solve with the quadratic formula

7 u^{2} + 14 u + 7 = 0

Quadratic Equation Formula:

For

a = 7, b = 14, c = 7

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

1 4^{2} - 4 \cdot 7 \cdot 7 = 0

1 4^{2} - 4 \cdot 7 \cdot 7

Multiply the numbers:

4 \cdot 7 \cdot 7 = 196

= 1 4^{2} - 196

1 4^{2} = 196

= 196 - 196

Subtract the numbers:

196 - 196 = 0

= 0

u_{1, 2} = \frac{- 14 \pm 0}{2 \cdot 7}

u = \frac{- 14}{2 \cdot 7}

\frac{- 14}{2 \cdot 7} = - 1

\frac{- 14}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 14}{14}

Apply the fraction rule:

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

= - \frac{14}{14}

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

tan (θ) = 17

tan (θ) = 14

cos (x) sin (x) = 4 sin (x)

cos (θ) = \frac{- 2}{2}

sin^{3} (x) = 0

What is the general solution for 7sin^2(x)+14sin(x)+7=0 ?
The general solution for 7sin^2(x)+14sin(x)+7=0 is x=(3pi)/2+2pin