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Popular Trigonometry >

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

Solution

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

Solution

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}, x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

Degrees

x = - 6 0^{\circ} - 48 0^{\circ} n, x = - 30 0^{\circ} - 48 0^{\circ} n, x = 6 0^{\circ} + 48 0^{\circ} n, x = 30 0^{\circ} + 48 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Rewrite using trig identities

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

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

Apply rule

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

cos (x) + cos (\frac{- x + π}{2}) = 0 : x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

cos (x) + cos (\frac{- x + π}{2}) = 0

Rewrite using trig identities

cos (\frac{- x + π}{2}) + cos (x)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

\frac{\frac{- x + π}{2} + x}{2}

Join

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

\frac{- x + π}{2} + x

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{- x + π}{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 + π + x \cdot 2}{2}

- x + π + x \cdot 2 = x + π

- x + π + x \cdot 2

Group like terms

= - x + 2 x + π

Add similar elements:

- x + 2 x = x

= x + π

= \frac{x + π}{2}

= \frac{\frac{x + π}{2}}{2}

Apply the fraction rule:

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

= \frac{x + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{x + π}{4}

= 2 cos (\frac{x + π}{4}) cos (\frac{\frac{- x + π}{2} - x}{2})

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

\frac{\frac{- x + π}{2} - x}{2}

Join

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

\frac{- x + π}{2} - x

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{- x + π}{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 + π - x \cdot 2}{2}

- x + π - x \cdot 2 = - 3 x + π

- x + π - x \cdot 2

Group like terms

= - x - 2 x + π

Add similar elements:

- x - 2 x = - 3 x

= - 3 x + π

= \frac{- 3 x + π}{2}

= \frac{\frac{- 3 x + π}{2}}{2}

Apply the fraction rule:

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

= \frac{- 3 x + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 x + π}{4}

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

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

2 cos (\frac{x + π}{4}) cos (\frac{π - 3 x}{4}) = 0

Solving each part separately

cos (\frac{x + π}{4}) = 0 or cos (\frac{π - 3 x}{4}) = 0

cos (\frac{x + π}{4}) = 0 : x = π + 8 πn, x = 5 π + 8 πn

cos (\frac{x + π}{4}) = 0

General solutions for

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

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

Solve

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

\frac{x + π}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{x + π}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( x + π )}{4} : x + π

\frac{4 ( x + π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= x + π

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 + π = 2 π + 8 πn

x + π = 2 π + 8 πn

x + π = 2 π + 8 πn

Move

π

to the right side

x + π = 2 π + 8 πn

Subtract

π

from both sides

x + π - π = 2 π + 8 πn - π

Simplify

x = π + 8 πn

x = π + 8 πn

Solve

\frac{x + π}{4} = \frac{3 π}{2} + 2 πn : x = 5 π + 8 πn

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( x + π )}{4} : x + π

\frac{4 ( x + π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= x + π

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 + π = 6 π + 8 πn

x + π = 6 π + 8 πn

x + π = 6 π + 8 πn

Move

π

to the right side

x + π = 6 π + 8 πn

Subtract

π

from both sides

x + π - π = 6 π + 8 πn - π

Simplify

x = 5 π + 8 πn

x = 5 π + 8 πn

x = π + 8 πn, x = 5 π + 8 πn

cos (\frac{π - 3 x}{4}) = 0 : x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

cos (\frac{π - 3 x}{4}) = 0

General solutions for

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

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

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( π - 3 x )}{4} : π - 3 x

\frac{4 ( π - 3 x )}{4}

Divide the numbers:

\frac{4}{4} = 1

= π - 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

π - 3 x = 2 π + 8 πn

π - 3 x = 2 π + 8 πn

π - 3 x = 2 π + 8 πn

Move

π

to the right side

π - 3 x = 2 π + 8 πn

Subtract

π

from both sides

π - 3 x - π = 2 π + 8 πn - π

Simplify

- 3 x = π + 8 πn

- 3 x = π + 8 πn

Divide both sides by

- 3

- 3 x = π + 8 πn

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{- 3} + \frac{8 πn}{- 3} : - \frac{π}{3} - \frac{8 πn}{3}

\frac{π}{- 3} + \frac{8 πn}{- 3}

Apply the fraction rule:

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

= - \frac{π}{3} + \frac{8 πn}{- 3}

Apply the fraction rule:

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

= - \frac{π}{3} - \frac{8 πn}{3}

x = - \frac{π}{3} - \frac{8 πn}{3}

x = - \frac{π}{3} - \frac{8 πn}{3}

x = - \frac{π}{3} - \frac{8 πn}{3}

Solve

\frac{π - 3 x}{4} = \frac{3 π}{2} + 2 πn : x = - \frac{5 π}{3} - \frac{8 πn}{3}

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( π - 3 x )}{4} : π - 3 x

\frac{4 ( π - 3 x )}{4}

Divide the numbers:

\frac{4}{4} = 1

= π - 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

π - 3 x = 6 π + 8 πn

π - 3 x = 6 π + 8 πn

π - 3 x = 6 π + 8 πn

Move

π

to the right side

π - 3 x = 6 π + 8 πn

Subtract

π

from both sides

π - 3 x - π = 6 π + 8 πn - π

Simplify

- 3 x = 5 π + 8 πn

- 3 x = 5 π + 8 πn

Divide both sides by

- 3

- 3 x = 5 π + 8 πn

Divide both sides by

- 3

\frac{- 3 x}{- 3} = \frac{5 π}{- 3} + \frac{8 πn}{- 3}

Simplify

\frac{- 3 x}{- 3} = \frac{5 π}{- 3} + \frac{8 πn}{- 3}

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{5 π}{- 3} + \frac{8 πn}{- 3} : - \frac{5 π}{3} - \frac{8 πn}{3}

\frac{5 π}{- 3} + \frac{8 πn}{- 3}

Apply the fraction rule:

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

= - \frac{5 π}{3} + \frac{8 πn}{- 3}

Apply the fraction rule:

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

= - \frac{5 π}{3} - \frac{8 πn}{3}

x = - \frac{5 π}{3} - \frac{8 πn}{3}

x = - \frac{5 π}{3} - \frac{8 πn}{3}

x = - \frac{5 π}{3} - \frac{8 πn}{3}

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

Combine all the solutions

x = π + 8 πn, x = 5 π + 8 πn, x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

Merge Overlapping Intervals

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

cos (x) - cos (\frac{- x + π}{2}) = 0 : x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

cos (x) - cos (\frac{- x + π}{2}) = 0

Rewrite using trig identities

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

Simplify

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

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

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

\frac{x + \frac{- x + π}{2}}{2}

Join

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

x + \frac{- x + π}{2}

Convert element to fraction:

x = \frac{x 2}{2}

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

Add similar elements:

2 x - x = x

= \frac{x + π}{2}

= \frac{\frac{x + π}{2}}{2}

Apply the fraction rule:

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

= \frac{x + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{x + π}{4}

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

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

\frac{x - \frac{- x + π}{2}}{2}

Join

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

x - \frac{- x + π}{2}

Convert element to fraction:

x = \frac{x 2}{2}

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

Expand

x \cdot 2 - (- x + π) : 3 x - π

x \cdot 2 - (- x + π)

= 2 x - (- x + π)

- (- x + π) : x - π

- (- x + π)

Distribute parentheses

= - (- x) - (π)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= x - π

= x \cdot 2 + x - π

Add similar elements:

2 x + x = 3 x

= 3 x - π

= \frac{3 x - π}{2}

= \frac{\frac{3 x - π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 x - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 x - π}{4}

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

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

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

Solving each part separately

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

sin (\frac{- π + 3 x}{4}) = 0 : x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

sin (\frac{- π + 3 x}{4}) = 0

General solutions for

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

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

Solve

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

\frac{- π + 3 x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{- π + 3 x}{4} = 2 πn

Multiply both sides by

4

\frac{- π + 3 x}{4} = 2 πn

Multiply both sides by

4

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

Simplify

- π + 3 x = 8 πn

- π + 3 x = 8 πn

Move

π

to the right side

- π + 3 x = 8 πn

Add

π

to both sides

- π + 3 x + π = 8 πn + π

Simplify

3 x = 8 πn + π

3 x = 8 πn + π

Divide both sides by

3

3 x = 8 πn + π

Divide both sides by

3

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

Simplify

x = \frac{8 πn}{3} + \frac{π}{3}

x = \frac{8 πn}{3} + \frac{π}{3}

Solve

\frac{- π + 3 x}{4} = π + 2 πn : x = \frac{5 π}{3} + \frac{8 πn}{3}

\frac{- π + 3 x}{4} = π + 2 πn

Multiply both sides by

4

\frac{- π + 3 x}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

- π + 3 x = 4 π + 8 πn

- π + 3 x = 4 π + 8 πn

Move

π

to the right side

- π + 3 x = 4 π + 8 πn

Add

π

to both sides

- π + 3 x + π = 4 π + 8 πn + π

Simplify

3 x = 5 π + 8 πn

3 x = 5 π + 8 πn

Divide both sides by

3

3 x = 5 π + 8 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{5 π}{3} + \frac{8 πn}{3}

Simplify

x = \frac{5 π}{3} + \frac{8 πn}{3}

x = \frac{5 π}{3} + \frac{8 πn}{3}

x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

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

sin (\frac{x + π}{4}) = 0

General solutions for

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

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

Solve

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

\frac{x + π}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x + π}{4} = 2 πn

Multiply both sides by

4

\frac{x + π}{4} = 2 πn

Multiply both sides by

4

\frac{4 ( x + π )}{4} = 4 \cdot 2 πn

Simplify

x + π = 8 πn

x + π = 8 πn

Move

π

to the right side

x + π = 8 πn

Subtract

π

from both sides

x + π - π = 8 πn - π

Simplify

x = 8 πn - π

x = 8 πn - π

Solve

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

\frac{x + π}{4} = π + 2 πn

Multiply both sides by

4

\frac{x + π}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

x + π = 4 π + 8 πn

x + π = 4 π + 8 πn

Move

π

to the right side

x + π = 4 π + 8 πn

Subtract

π

from both sides

x + π - π = 4 π + 8 πn - π

Simplify

x = 3 π + 8 πn

x = 3 π + 8 πn

x = 8 πn - π, x = 3 π + 8 πn

Combine all the solutions

x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}, x = 8 πn - π, x = 3 π + 8 πn

Merge Overlapping Intervals

x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

Combine all the solutions

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}, x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

Graph

Popular Examples

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