What is the general solution for cos(2x)=-1/2 ?

The general solution for cos(2x)=-1/2 is x= pi/3+pin,x=(2pi)/3+pin

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cos(2x)=-1/2

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

General solutions for

cos (2 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}

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

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

Solve

2 x = \frac{2 π}{3} + 2 πn : x = \frac{π}{3} + πn

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{3} + πn

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

Solve

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

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{4 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2} : \frac{2 π}{3} + πn

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{4 π}{3}}{2} = \frac{2 π}{3}

\frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

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

Popular Examples

cos(x)=5

cos (x) = 5

2cos(θ)-sqrt(3)=0

2 cos (θ) - 3 = 0

tan(pi)

tan (π)

cos^2(60)

cos^{2} (6 0^{\circ})

sin(2θ)=(sqrt(3))/2

sin (2 θ) = \frac{3}{2}

Frequently Asked Questions (FAQ)

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