What is the general solution for sin^2(x)+5sin(x)+5=0 ?

The general solution for sin^2(x)+5sin(x)+5=0 is No Solution for x\in\mathbb{R}

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sin^2(x)+5sin(x)+5=0

sin^{2} (x) + 5 sin (x) + 5 = 0

No Solution for x \in R

Solution steps

sin^{2} (x) + 5 sin (x) + 5 = 0

Solve by substitution

sin^{2} (x) + 5 sin (x) + 5 = 0

Let:

sin (x) = u

u^{2} + 5 u + 5 = 0

u^{2} + 5 u + 5 = 0 : u = \frac{- 5 + 5}{2}, u = \frac{- 5 - 5}{2}

u^{2} + 5 u + 5 = 0

Solve with the quadratic formula

u^{2} + 5 u + 5 = 0

Quadratic Equation Formula:

For

a = 1, b = 5, c = 5

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot 5}{2 \cdot 1}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot 5}{2 \cdot 1}

5^{2} - 4 \cdot 1 \cdot 5 = 5

5^{2} - 4 \cdot 1 \cdot 5

Multiply the numbers:

4 \cdot 1 \cdot 5 = 20

= 5^{2} - 20

5^{2} = 25

= 25 - 20

Subtract the numbers:

25 - 20 = 5

= 5

u_{1, 2} = \frac{- 5 \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 5 + 5}{2 \cdot 1}, u_{2} = \frac{- 5 - 5}{2 \cdot 1}

u = \frac{- 5 + 5}{2 \cdot 1} : \frac{- 5 + 5}{2}

\frac{- 5 + 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 5 + 5}{2}

u = \frac{- 5 - 5}{2 \cdot 1} : \frac{- 5 - 5}{2}

\frac{- 5 - 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 5 - 5}{2}

The solutions to the quadratic equation are:

u = \frac{- 5 + 5}{2}, u = \frac{- 5 - 5}{2}

Substitute back

u = sin (x)

sin (x) = \frac{- 5 + 5}{2}, sin (x) = \frac{- 5 - 5}{2}

sin (x) = \frac{- 5 + 5}{2}, sin (x) = \frac{- 5 - 5}{2}

sin (x) = \frac{- 5 + 5}{2} :

No Solution

sin (x) = \frac{- 5 + 5}{2}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{- 5 - 5}{2} :

No Solution

sin (x) = \frac{- 5 - 5}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

cos (x) = - 0.814

3 cos^{2} (x) - 8 cos (x) + 4 = sin^{2} (x)

cos (x - 4 5^{\circ}) = \frac{- 1}{2}

x, sin (x + \frac{r}{4}) + sin (x - \frac{r}{3}) = 0

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

What is the general solution for sin^2(x)+5sin(x)+5=0 ?
The general solution for sin^2(x)+5sin(x)+5=0 is No Solution for x\in\mathbb{R}