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

Q: What is the general solution for 2sin^2(x)+sin(x)=1,-pi<= x<= pi ?

The general solution for 2sin^2(x)+sin(x)=1,-pi<= x<= pi is x= pi/6 ,x=(5pi)/6 ,x=-pi/2

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2sin^2(x)+sin(x)=1,-pi<= x<= pi

Solution

2 sin^{2} (x) + sin (x) = 1, - π \leq x \leq π

Solution

x = \frac{π}{6}, x = \frac{5 π}{6}, x = - \frac{π}{2}

Degrees

x = 3 0^{\circ}, x = 15 0^{\circ}, x = - 9 0^{\circ}

Solution steps

2 sin^{2} (x) + sin (x) = 1, - π \leq x \leq π

Solve by substitution

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

Let:

sin (x) = u

2 u^{2} + u = 1

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

2 u^{2} + u = 1

Move

1

to the left side

2 u^{2} + u = 1

Subtract

1

from both sides

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

Simplify

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

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 (x)

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

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

sin (x) = \frac{1}{2}, - π \leq x \leq π : x = \frac{π}{6}, x = \frac{5 π}{6}

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

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

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

Solutions for the range

- π \leq x \leq π

x = \frac{π}{6}, x = \frac{5 π}{6}

sin (x) = - 1, - π \leq x \leq π : x = - \frac{π}{2}

sin (x) = - 1, - π \leq x \leq π

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

Solutions for the range

- π \leq x \leq π

x = - \frac{π}{2}

Combine all the solutions

x = \frac{π}{6}, x = \frac{5 π}{6}, x = - \frac{π}{2}

Popular Examples

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1 + tan^{2} (x) = cot^{2} (x)

1-4tan(3x+1)=17

1 - 4 tan (3 x + 1) = 17

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

2-2sin^2(x)=-3cos(x)+2

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

Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)+sin(x)=1,-pi<= x<= pi ?
The general solution for 2sin^2(x)+sin(x)=1,-pi<= x<= pi is x= pi/6 ,x=(5pi)/6 ,x=-pi/2