What is the general solution for cos(x)-cos(x/2)=0 ?

The general solution for cos(x)-cos(x/2)=0 is x=(8pin)/3 ,x=(4pi)/3+(8pin)/3

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cos(x)-cos(x/2)=0

Solution

cos (x) - cos (\frac{x}{2}) = 0

Solution

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

Degrees

x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n

Solution steps

cos (x) - cos (\frac{x}{2}) = 0

Rewrite using trig identities

- cos (\frac{x}{2}) + cos (x)

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{x + \frac{x}{2}}{2}) sin (\frac{x - \frac{x}{2}}{2})

Simplify

- 2 sin (\frac{x + \frac{x}{2}}{2}) sin (\frac{x - \frac{x}{2}}{2}) : - 2 sin (\frac{3 x}{4}) sin (\frac{x}{4})

- 2 sin (\frac{x + \frac{x}{2}}{2}) sin (\frac{x - \frac{x}{2}}{2})

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

\frac{x + \frac{x}{2}}{2}

Join

x + \frac{x}{2} : \frac{3 x}{2}

x + \frac{x}{2}

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{x \cdot 2}{2} + \frac{x}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 2 + x}{2}

Add similar elements:

2 x + x = 3 x

= \frac{3 x}{2}

= \frac{\frac{3 x}{2}}{2}

Apply the fraction rule:

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

= \frac{3 x}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 x}{4}

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

\frac{x - \frac{x}{2}}{2} = \frac{x}{4}

\frac{x - \frac{x}{2}}{2}

Join

x - \frac{x}{2} : \frac{x}{2}

x - \frac{x}{2}

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{x \cdot 2}{2} - \frac{x}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 2 - x}{2}

Add similar elements:

2 x - x = x

= \frac{x}{2}

= \frac{\frac{x}{2}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{x}{4}

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

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

- 2 sin (\frac{3 x}{4}) sin (\frac{x}{4}) = 0

Solving each part separately

sin (\frac{3 x}{4}) = 0 or sin (\frac{x}{4}) = 0

sin (\frac{3 x}{4}) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (\frac{3 x}{4}) = 0

General solutions for

sin (\frac{3 x}{4}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

\frac{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

Solve

\frac{3 x}{4} = 0 + 2 πn : x = \frac{8 πn}{3}

\frac{3 x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{4} = 2 πn

Multiply both sides by

4

\frac{3 x}{4} = 2 πn

Multiply both sides by

4

\frac{4 \cdot 3 x}{4} = 4 \cdot 2 πn

Simplify

3 x = 8 πn

3 x = 8 πn

Divide both sides by

3

3 x = 8 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{8 πn}{3}

Simplify

x = \frac{8 πn}{3}

x = \frac{8 πn}{3}

Solve

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

\frac{3 x}{4} = π + 2 πn

Multiply both sides by

4

\frac{3 x}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

3 x = 4 π + 8 πn

3 x = 4 π + 8 πn

Divide both sides by

3

3 x = 4 π + 8 πn

Divide both sides by

3

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

Simplify

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

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

sin (\frac{x}{4}) = 0 : x = 8 πn, x = 4 π + 8 πn

sin (\frac{x}{4}) = 0

General solutions for

sin (\frac{x}{4}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

Solve

\frac{x}{4} = 0 + 2 πn : x = 8 πn

\frac{x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{4} = 2 πn

Multiply both sides by

4

\frac{x}{4} = 2 πn

Multiply both sides by

4

\frac{4 x}{4} = 4 \cdot 2 πn

Simplify

x = 8 πn

x = 8 πn

Solve

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

\frac{x}{4} = π + 2 πn

Multiply both sides by

4

\frac{x}{4} = π + 2 πn

Multiply both sides by

4

\frac{4 x}{4} = 4 π + 4 \cdot 2 πn

Simplify

x = 4 π + 8 πn

x = 4 π + 8 πn

x = 8 πn, x = 4 π + 8 πn

Combine all the solutions

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

Merge Overlapping Intervals

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)-cos(x/2)=0 ?
The general solution for cos(x)-cos(x/2)=0 is x=(8pin)/3 ,x=(4pi)/3+(8pin)/3