What is the general solution for 1/2 cos(3x)=-1/4 ?

The general solution for 1/2 cos(3x)=-1/4 is x=(2pi)/9+(2pin)/3 ,x=(4pi)/9+(2pin)/3

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1/2 cos(3x)=-1/4

Solution

\frac{1}{2} cos (3 x) = - \frac{1}{4}

Solution

x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Degrees

x = 4 0^{\circ} + 12 0^{\circ} n, x = 8 0^{\circ} + 12 0^{\circ} n

Solution steps

\frac{1}{2} cos (3 x) = - \frac{1}{4}

Multiply both sides by

2

\frac{1}{2} cos (3 x) = - \frac{1}{4}

Multiply both sides by

2

2 \cdot \frac{1}{2} cos (3 x) = 2 (- \frac{1}{4})

Simplify

2 \cdot \frac{1}{2} cos (3 x) = 2 (- \frac{1}{4})

Simplify

2 \cdot \frac{1}{2} cos (3 x) : cos (3 x)

2 \cdot \frac{1}{2} cos (3 x)

Multiply fractions:

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

= \frac{1 \cdot 2}{2} cos (3 x)

Cancel the common factor:

2

= cos (3 x) \cdot 1

Multiply:

cos (3 x) \cdot 1 = cos (3 x)

= cos (3 x)

Simplify

2 (- \frac{1}{4}) : - \frac{1}{2}

2 (- \frac{1}{4})

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

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

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

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

General solutions for

cos (3 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}

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

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

Solve

3 x = \frac{2 π}{3} + 2 πn : x = \frac{2 π}{9} + \frac{2 πn}{3}

3 x = \frac{2 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{2 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{3} + \frac{2 πn}{3} : \frac{2 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{2 π}{3}}{3} = \frac{2 π}{9}

\frac{\frac{2 π}{3}}{3}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 π}{9}

= \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

Solve

3 x = \frac{4 π}{3} + 2 πn : x = \frac{4 π}{9} + \frac{2 πn}{3}

3 x = \frac{4 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{4 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{4 π}{3}}{3} + \frac{2 πn}{3} : \frac{4 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{4 π}{3}}{3} = \frac{4 π}{9}

\frac{\frac{4 π}{3}}{3}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{4 π}{9}

= \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Popular Examples

cos(x)= 4/5 ,csc(x)<0

cos (x) = \frac{4}{5}, csc (x) < 0

cos(y)=0.5736

cos (y) = 0.5736

cot(θ)=-6/5

cot (θ) = - \frac{6}{5}

cot(x)cos(x)-cot(x)=0

cot (x) cos (x) - cot (x) = 0

sec^2(θ)-25=0

sec^{2} (θ) - 25 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 1/2 cos(3x)=-1/4 ?
The general solution for 1/2 cos(3x)=-1/4 is x=(2pi)/9+(2pin)/3 ,x=(4pi)/9+(2pin)/3