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

The general solution for 3cos(x/2)+sqrt(3)=cos(x/2) is x=(5pi)/3+4pin,x=(7pi)/3+4pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3cos(x/2)+sqrt(3)=cos(x/2)

Solution

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

Solution

x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

Degrees

x = 30 0^{\circ} + 72 0^{\circ} n, x = 42 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (\frac{x}{2}) = u

3 u + 3 = u

3 u + 3 = u : u = - \frac{3}{2}

3 u + 3 = u

Move

3

to the right side

3 u + 3 = u

Subtract

3

from both sides

3 u + 3 - 3 = u - 3

Simplify

3 u = u - 3

3 u = u - 3

Move

u

to the left side

3 u = u - 3

Subtract

u

from both sides

3 u - u = u - 3 - u

Simplify

2 u = - 3

2 u = - 3

Divide both sides by

2

2 u = - 3

Divide both sides by

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

Substitute back

u = cos (\frac{x}{2})

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

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

cos (\frac{x}{2}) = - \frac{3}{2} : x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

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

General solutions for

cos (\frac{x}{2}) = - \frac{3}{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{x}{2} = \frac{5 π}{6} + 2 πn, \frac{x}{2} = \frac{7 π}{6} + 2 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn, \frac{x}{2} = \frac{7 π}{6} + 2 πn

Solve

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

Solve

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

Combine all the solutions

x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

Popular Examples

(tan(θ)+1)(tan(θ)-sqrt(3))=0

(tan (θ) + 1) (tan (θ) - 3) = 0

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

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

2cos^2(x)-2sin^2(x)=0

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

csc^2(θ)+5=7-csc(θ)

csc^{2} (θ) + 5 = 7 - csc (θ)

3tan^2(x/3)-1=0

3 tan^{2} (\frac{x}{3}) - 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 3cos(x/2)+sqrt(3)=cos(x/2) ?
The general solution for 3cos(x/2)+sqrt(3)=cos(x/2) is x=(5pi)/3+4pin,x=(7pi)/3+4pin