What is the general solution for 4sin^2(x)+4sin(x)=-1 ?

The general solution for 4sin^2(x)+4sin(x)=-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

4sin^2(x)+4sin(x)=-1

Solution

4 sin^{2} (x) + 4 sin (x) = - 1

Solution

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

Degrees

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

Solution steps

4 sin^{2} (x) + 4 sin (x) = - 1

Solve by substitution

4 sin^{2} (x) + 4 sin (x) = - 1

Let:

sin (x) = u

4 u^{2} + 4 u = - 1

4 u^{2} + 4 u = - 1 : u = - \frac{1}{2}

4 u^{2} + 4 u = - 1

Move

1

to the left side

4 u^{2} + 4 u = - 1

Add

1

to both sides

4 u^{2} + 4 u + 1 = - 1 + 1

Simplify

4 u^{2} + 4 u + 1 = 0

4 u^{2} + 4 u + 1 = 0

Solve with the quadratic formula

4 u^{2} + 4 u + 1 = 0

Quadratic Equation Formula:

For

a = 4, b = 4, c = 1

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

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

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

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

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

u = \frac{- 4}{2 \cdot 4}

\frac{- 4}{2 \cdot 4} = - \frac{1}{2}

\frac{- 4}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

u = - \frac{1}{2}

The solution to the quadratic equation is:

u = - \frac{1}{2}

Substitute back

u = sin (x)

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

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

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

Combine all the solutions

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

Graph

Popular Examples

sin(θ)= 8/17 ,cos(θ)

sin (θ) = \frac{8}{17}, cos (θ)

solvefor θ,3cos(θ-55)=-1.56 solve for

θ, 3 cos (θ - 5 5^{\circ}) = - 1.56

-6sin(c)-12=2sin(c)-4

- 6 sin (c) - 12 = 2 sin (c) - 4

solvefor x,3cot^2(x)-1=0 solve for

x, 3 cot^{2} (x) - 1 = 0

tan(2x+10)=cot(3x-15)

tan (2 x + 10) = cot (3 x - 15)

Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x)+4sin(x)=-1 ?
The general solution for 4sin^2(x)+4sin(x)=-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin