What is the general solution for 2sin^2(w)+sin(w)-1=0 ?

The general solution for 2sin^2(w)+sin(w)-1=0 is w= pi/6+2pin,w=(5pi)/6+2pin,w=(3pi)/2+2pin

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2sin^2(w)+sin(w)-1=0

Solution

2 sin^{2} (w) + sin (w) - 1 = 0

Solution

w = \frac{π}{6} + 2 πn, w = \frac{5 π}{6} + 2 πn, w = \frac{3 π}{2} + 2 πn

Degrees

w = 3 0^{\circ} + 36 0^{\circ} n, w = 15 0^{\circ} + 36 0^{\circ} n, w = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (w) + sin (w) - 1 = 0

Solve by substitution

2 sin^{2} (w) + sin (w) - 1 = 0

Let:

sin (w) = u

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

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 2, b = 1, c = - 1

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

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

1^{2} - 4 \cdot 2 (- 1) = 3

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 1)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

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

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

\frac{- 1 + 3}{2 \cdot 2}

Add/Subtract the numbers:

- 1 + 3 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

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

\frac{- 1 - 3}{2 \cdot 2}

Subtract the numbers:

- 1 - 3 = - 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

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

Substitute back

u = sin (w)

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

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

sin (w) = \frac{1}{2} : w = \frac{π}{6} + 2 πn, w = \frac{5 π}{6} + 2 πn

sin (w) = \frac{1}{2}

General solutions for

sin (w) = \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}

w = \frac{π}{6} + 2 πn, w = \frac{5 π}{6} + 2 πn

w = \frac{π}{6} + 2 πn, w = \frac{5 π}{6} + 2 πn

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

sin (w) = - 1

General solutions for

sin (w) = - 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}

w = \frac{3 π}{2} + 2 πn

w = \frac{3 π}{2} + 2 πn

Combine all the solutions

w = \frac{π}{6} + 2 πn, w = \frac{5 π}{6} + 2 πn, w = \frac{3 π}{2} + 2 πn

Popular Examples

sin^2(x)+sin(x)=-cos^2(x)

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

arctan(x)=arccos(3/4)

arctan (x) = arccos (\frac{3}{4})

tan^2(θ)+7tan(θ)+9=0

tan^{2} (θ) + 7 tan (θ) + 9 = 0

solvefor x,cos(x)*cos(y)=0 solve for

x, cos (x) \cdot cos (y) = 0

3sec(θ)=6

3 sec (θ) = 6

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(w)+sin(w)-1=0 ?
The general solution for 2sin^2(w)+sin(w)-1=0 is w= pi/6+2pin,w=(5pi)/6+2pin,w=(3pi)/2+2pin