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

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

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

Solution

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

Solution

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

Degrees

x = 205.94366 \dots^{\circ} + 36 0^{\circ} n, x = 325.94366 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

General solutions for

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

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 2 x - 3 )}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - 3 )}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - 3 )}{2} : 2 x - 3

\frac{2 ( 2 x - 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - 3

Simplify

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

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

2 x - 3 = \frac{4 π}{3} + 4 πn

2 x - 3 = \frac{4 π}{3} + 4 πn

2 x - 3 = \frac{4 π}{3} + 4 πn

Move

3

to the right side

2 x - 3 = \frac{4 π}{3} + 4 πn

Add

3

to both sides

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

Simplify

2 x = \frac{4 π}{3} + 4 πn + 3

2 x = \frac{4 π}{3} + 4 πn + 3

Divide both sides by

2

2 x = \frac{4 π}{3} + 4 πn + 3

Divide both sides by

2

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

Simplify

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

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

Group like terms

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

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\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{3}{2} + 2 πn + \frac{2 π}{3}

Group like terms

= 2 πn + \frac{3}{2} + \frac{2 π}{3}

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

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 2 x - 3 )}{2} = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - 3 )}{2} = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - 3 )}{2} : 2 x - 3

\frac{2 ( 2 x - 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - 3

Simplify

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn : \frac{8 π}{3} + 4 πn

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{4 π}{3} = \frac{8 π}{3}

2 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 2}{3}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{8 π}{3} + 4 πn

2 x - 3 = \frac{8 π}{3} + 4 πn

2 x - 3 = \frac{8 π}{3} + 4 πn

2 x - 3 = \frac{8 π}{3} + 4 πn

Move

3

to the right side

2 x - 3 = \frac{8 π}{3} + 4 πn

Add

3

to both sides

2 x - 3 + 3 = \frac{8 π}{3} + 4 πn + 3

Simplify

2 x = \frac{8 π}{3} + 4 πn + 3

2 x = \frac{8 π}{3} + 4 πn + 3

Divide both sides by

2

2 x = \frac{8 π}{3} + 4 πn + 3

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{8 π}{3}}{2} + \frac{4 πn}{2} + \frac{3}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{8 π}{3}}{2} + \frac{4 πn}{2} + \frac{3}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{8 π}{3}}{2} + \frac{4 πn}{2} + \frac{3}{2} : 2 πn + \frac{3}{2} + \frac{4 π}{3}

\frac{\frac{8 π}{3}}{2} + \frac{4 πn}{2} + \frac{3}{2}

Group like terms

= \frac{3}{2} + \frac{4 πn}{2} + \frac{\frac{8 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{8 π}{3}}{2} = \frac{4 π}{3}

\frac{\frac{8 π}{3}}{2}

Apply the fraction rule:

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

= \frac{8 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{8 π}{6}

Cancel the common factor:

2

= \frac{4 π}{3}

= \frac{3}{2} + 2 πn + \frac{4 π}{3}

Group like terms

= 2 πn + \frac{3}{2} + \frac{4 π}{3}

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

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

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

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

Graph

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Frequently Asked Questions (FAQ)

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