What is the general solution for 2cos^2(x)-cos(x)-1=0,(0,2pi) ?

The general solution for 2cos^2(x)-cos(x)-1=0,(0,2pi) is x=(2pi)/3 ,x=(4pi)/3

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2cos^2(x)-cos(x)-1=0,(0,2pi)

Solution

2 cos^{2} (x) - cos (x) - 1 = 0, (0, 2 π)

Solution

x = \frac{2 π}{3}, x = \frac{4 π}{3}

Degrees

x = 12 0^{\circ}, x = 24 0^{\circ}

Solution steps

2 cos^{2} (x) - cos (x) - 1 = 0, (0, 2 π)

Solve by substitution

2 cos^{2} (x) - cos (x) - 1 = 0

Let:

cos (x) = u

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

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

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

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 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} : 1

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

Apply rule

- (- a) = a

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

Add the numbers:

1 + 3 = 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 3 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = 1, (0, 2 π) :

No Solution

cos (x) = 1, (0, 2 π)

General solutions for

cos (x) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Solutions for the range

(0, 2 π)

No Solution

cos (x) = - \frac{1}{2}, (0, 2 π) : x = \frac{2 π}{3}, x = \frac{4 π}{3}

cos (x) = - \frac{1}{2}, (0, 2 π)

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Solutions for the range

(0, 2 π)

x = \frac{2 π}{3}, x = \frac{4 π}{3}

Combine all the solutions

x = \frac{2 π}{3}, x = \frac{4 π}{3}

Popular Examples

-3+cos(2x)*2=0

- 3 + cos (2 x) \cdot 2 = 0

1/9+cos^2(x)=1

\frac{1}{9} + cos^{2} (x) = 1

sin(x)=0.9063

sin (x) = 0.9063

11=12+3.5sin((2pi)/(365)t)

11 = 12 + 3.5 sin (\frac{2 π}{365} t)

sin(2t)=2sin(t)

sin (2 t) = 2 sin (t)

Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(x)-cos(x)-1=0,(0,2pi) ?
The general solution for 2cos^2(x)-cos(x)-1=0,(0,2pi) is x=(2pi)/3 ,x=(4pi)/3