What is the general solution for 5/2 =2+cos(x+(2pi)/3) ?

The general solution for 5/2 =2+cos(x+(2pi)/3) is x=2pin-pi/3 ,x=2pin+pi

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Popular Trigonometry >

5/2 =2+cos(x+(2pi)/3)

Solution

\frac{5}{2} = 2 + cos (x + \frac{2 π}{3})

Solution

x = 2 πn - \frac{π}{3}, x = 2 πn + π

Degrees

x = - 6 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{5}{2} = 2 + cos (x + \frac{2 π}{3})

Switch sides

2 + cos (x + \frac{2 π}{3}) = \frac{5}{2}

Move

2

to the right side

2 + cos (x + \frac{2 π}{3}) = \frac{5}{2}

Subtract

2

from both sides

2 + cos (x + \frac{2 π}{3}) - 2 = \frac{5}{2} - 2

Simplify

2 + cos (x + \frac{2 π}{3}) - 2 = \frac{5}{2} - 2

Simplify

2 + cos (x + \frac{2 π}{3}) - 2 : cos (x + \frac{2 π}{3})

2 + cos (x + \frac{2 π}{3}) - 2

Add similar elements:

2 - 2 = 0

= cos (x + \frac{2 π}{3})

Simplify

\frac{5}{2} - 2 : \frac{1}{2}

\frac{5}{2} - 2

Convert element to fraction:

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

= - \frac{2 \cdot 2}{2} + \frac{5}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 \cdot 2 + 5}{2}

- 2 \cdot 2 + 5 = 1

- 2 \cdot 2 + 5

Multiply the numbers:

2 \cdot 2 = 4

= - 4 + 5

Add/Subtract the numbers:

- 4 + 5 = 1

= 1

= \frac{1}{2}

cos (x + \frac{2 π}{3}) = \frac{1}{2}

cos (x + \frac{2 π}{3}) = \frac{1}{2}

cos (x + \frac{2 π}{3}) = \frac{1}{2}

General solutions for

cos (x + \frac{2 π}{3}) = \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} = \frac{π}{3} + 2 πn, x + \frac{2 π}{3} = \frac{5 π}{3} + 2 πn

x + \frac{2 π}{3} = \frac{π}{3} + 2 πn, x + \frac{2 π}{3} = \frac{5 π}{3} + 2 πn

Solve

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

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

Move

\frac{2 π}{3}

to the right side

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

Subtract

\frac{2 π}{3}

from both sides

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

Simplify

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

Simplify

x + \frac{2 π}{3} - \frac{2 π}{3} : x

x + \frac{2 π}{3} - \frac{2 π}{3}

Add similar elements:

\frac{2 π}{3} - \frac{2 π}{3} = 0

= x

Simplify

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

\frac{π}{3} + 2 πn - \frac{2 π}{3}

Group like terms

= 2 πn + \frac{π}{3} - \frac{2 π}{3}

Combine the fractions

\frac{π}{3} - \frac{2 π}{3} : - \frac{π}{3}

Apply rule

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

= \frac{π - 2 π}{3}

Add similar elements:

π - 2 π = - π

= \frac{- π}{3}

Apply the fraction rule:

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

= - \frac{π}{3}

= 2 πn - \frac{π}{3}

x = 2 πn - \frac{π}{3}

x = 2 πn - \frac{π}{3}

x = 2 πn - \frac{π}{3}

Solve

x + \frac{2 π}{3} = \frac{5 π}{3} + 2 πn : x = 2 πn + π

x + \frac{2 π}{3} = \frac{5 π}{3} + 2 πn

Move

\frac{2 π}{3}

to the right side

x + \frac{2 π}{3} = \frac{5 π}{3} + 2 πn

Subtract

\frac{2 π}{3}

from both sides

x + \frac{2 π}{3} - \frac{2 π}{3} = \frac{5 π}{3} + 2 πn - \frac{2 π}{3}

Simplify

x + \frac{2 π}{3} - \frac{2 π}{3} = \frac{5 π}{3} + 2 πn - \frac{2 π}{3}

Simplify

x + \frac{2 π}{3} - \frac{2 π}{3} : x

x + \frac{2 π}{3} - \frac{2 π}{3}

Add similar elements:

\frac{2 π}{3} - \frac{2 π}{3} = 0

= x

Simplify

\frac{5 π}{3} + 2 πn - \frac{2 π}{3} : 2 πn + π

\frac{5 π}{3} + 2 πn - \frac{2 π}{3}

Group like terms

= 2 πn + \frac{5 π}{3} - \frac{2 π}{3}

Combine the fractions

\frac{5 π}{3} - \frac{2 π}{3} : π

Apply rule

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

= \frac{5 π - 2 π}{3}

Add similar elements:

5 π - 2 π = 3 π

= \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn - \frac{π}{3}, x = 2 πn + π

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5/2 =2+cos(x+(2pi)/3) ?
The general solution for 5/2 =2+cos(x+(2pi)/3) is x=2pin-pi/3 ,x=2pin+pi