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Popular Trigonometry >

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

Solution

2 - 2 cos^{2} (x) = 2 sin^{2} (\frac{x}{2})

Solution

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

Degrees

x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n, x = 12 0^{\circ} + 48 0^{\circ} n, x = 36 0^{\circ} + 48 0^{\circ} n

Solution steps

2 - 2 cos^{2} (x) = 2 sin^{2} (\frac{x}{2})

Subtract

2 sin^{2} (\frac{x}{2})

from both sides

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

Rewrite using trig identities

2 - 2 cos^{2} (x) - 2 sin^{2} (\frac{x}{2})

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 2 - 2 (1 - sin^{2} (x)) - 2 sin^{2} (\frac{x}{2})

Simplify

2 - 2 (1 - sin^{2} (x)) - 2 sin^{2} (\frac{x}{2}) : 2 sin^{2} (x) - 2 sin^{2} (\frac{x}{2})

2 - 2 (1 - sin^{2} (x)) - 2 sin^{2} (\frac{x}{2})

Expand

- 2 (1 - sin^{2} (x)) : - 2 + 2 sin^{2} (x)

- 2 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = sin^{2} (x)

= - 2 \cdot 1 - (- 2) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 sin^{2} (x)

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

2 - 2 = 0

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

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

- 2 sin^{2} (\frac{x}{2}) + 2 sin^{2} (x) = 0

Factor

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

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

Factor out common term

2

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

Factor

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

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

Solving each part separately

sin (x) + sin (\frac{x}{2}) = 0 or sin (x) - sin (\frac{x}{2}) = 0

sin (x) + sin (\frac{x}{2}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (x) + sin (\frac{x}{2}) = 0

Rewrite using trig identities

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

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

Join

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

\frac{x}{2} + x

Convert element to fraction:

x = \frac{x 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

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

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

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

Join

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

\frac{x}{2} - x

Convert element to fraction:

x = \frac{x 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{x}{4}

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

Use the negative angle identity:

cos (- x) = cos (x)

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

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

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

Solving each part separately

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

cos (\frac{x}{4}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn

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

General solutions for

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

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

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

Solve

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

\frac{x}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

Solve

\frac{x}{4} = \frac{3 π}{2} + 2 πn : x = 6 π + 8 πn

\frac{x}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 2 π + 8 πn, x = 6 π + 8 πn

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}

Combine all the solutions

x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (x) - sin (\frac{x}{2}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

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

Rewrite using trig identities

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

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

Solving each part separately

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

cos (\frac{3 x}{4}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

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

General solutions for

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

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

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

Solve

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

\frac{3 x}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{3 x}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 \cdot 3 x}{4} : 3 x

\frac{4 \cdot 3 x}{4}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{12 x}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 x

Simplify

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

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

3 x = 2 π + 8 πn

3 x = 2 π + 8 πn

3 x = 2 π + 8 πn

Divide both sides by

3

3 x = 2 π + 8 πn

Divide both sides by

3

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

Simplify

x = \frac{2 π}{3} + \frac{8 πn}{3}

x = \frac{2 π}{3} + \frac{8 πn}{3}

Solve

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

\frac{3 x}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{3 x}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 \cdot 3 x}{4} : 3 x

\frac{4 \cdot 3 x}{4}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{12 x}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 x

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

3 x = 6 π + 8 πn

3 x = 6 π + 8 πn

3 x = 6 π + 8 πn

Divide both sides by

3

3 x = 6 π + 8 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{6 π}{3} + \frac{8 πn}{3}

Simplify

x = 2 π + \frac{8 πn}{3}

x = 2 π + \frac{8 πn}{3}

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

Combine all the solutions

x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}, x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \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}, x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

Graph

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