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2sin^2(θ/2)-2cos^2(θ)=0

Solution

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

Solution

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

Degrees

θ = - 6 0^{\circ} - 48 0^{\circ} n, θ = - 30 0^{\circ} - 48 0^{\circ} n, θ = 6 0^{\circ} - 48 0^{\circ} n, θ = - 18 0^{\circ} - 48 0^{\circ} n

Solution steps

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

Rewrite using trig identities

- 2 cos^{2} (θ) + 2 sin^{2} (\frac{θ}{2})

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

Apply rule

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

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

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

Factor

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

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

Factor out common term

2

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

Solving each part separately

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

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

cos (\frac{- θ + π}{2}) + cos (θ) = 0

Rewrite using trig identities

cos (\frac{- θ + π}{2}) + cos (θ)

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

Simplify

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

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

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

\frac{\frac{- θ + π}{2} + θ}{2}

Join

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

\frac{- θ + π}{2} + θ

Convert element to fraction:

θ = \frac{θ 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{- θ + π + θ \cdot 2}{2}

- θ + π + θ \cdot 2 = θ + π

- θ + π + θ \cdot 2

Group like terms

= - θ + 2 θ + π

Add similar elements:

- θ + 2 θ = θ

= θ + π

= \frac{θ + π}{2}

= \frac{\frac{θ + π}{2}}{2}

Apply the fraction rule:

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

= \frac{θ + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{θ + π}{4}

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

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

\frac{\frac{- θ + π}{2} - θ}{2}

Join

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

\frac{- θ + π}{2} - θ

Convert element to fraction:

θ = \frac{θ 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{- θ + π - θ \cdot 2}{2}

- θ + π - θ \cdot 2 = - 3 θ + π

- θ + π - θ \cdot 2

Group like terms

= - θ - 2 θ + π

Add similar elements:

- θ - 2 θ = - 3 θ

= - 3 θ + π

= \frac{- 3 θ + π}{2}

= \frac{\frac{- 3 θ + π}{2}}{2}

Apply the fraction rule:

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

= \frac{- 3 θ + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 θ + π}{4}

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

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

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

Solving each part separately

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

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

cos (\frac{θ + π}{4}) = 0

General solutions for

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

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

Solve

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

\frac{θ + π}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{θ + π}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( θ + π )}{4} : θ + π

\frac{4 ( θ + π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ + 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

θ + π = 2 π + 8 πn

θ + π = 2 π + 8 πn

θ + π = 2 π + 8 πn

Move

π

to the right side

θ + π = 2 π + 8 πn

Subtract

π

from both sides

θ + π - π = 2 π + 8 πn - π

Simplify

θ = π + 8 πn

θ = π + 8 πn

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( θ + π )}{4} : θ + π

\frac{4 ( θ + π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ + 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

θ + π = 6 π + 8 πn

θ + π = 6 π + 8 πn

θ + π = 6 π + 8 πn

Move

π

to the right side

θ + π = 6 π + 8 πn

Subtract

π

from both sides

θ + π - π = 6 π + 8 πn - π

Simplify

θ = 5 π + 8 πn

θ = 5 π + 8 πn

θ = π + 8 πn, θ = 5 π + 8 πn

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

cos (\frac{π - 3 θ}{4}) = 0

General solutions for

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

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

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( π - 3 θ )}{4} : π - 3 θ

\frac{4 ( π - 3 θ )}{4}

Divide the numbers:

\frac{4}{4} = 1

= π - 3 θ

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 θ = 2 π + 8 πn

π - 3 θ = 2 π + 8 πn

π - 3 θ = 2 π + 8 πn

Move

π

to the right side

π - 3 θ = 2 π + 8 πn

Subtract

π

from both sides

π - 3 θ - π = 2 π + 8 πn - π

Simplify

- 3 θ = π + 8 πn

- 3 θ = π + 8 πn

Divide both sides by

- 3

- 3 θ = π + 8 πn

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 θ}{- 3} : θ

\frac{- 3 θ}{- 3}

Apply the fraction rule:

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

= \frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

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}

θ = - \frac{π}{3} - \frac{8 πn}{3}

θ = - \frac{π}{3} - \frac{8 πn}{3}

θ = - \frac{π}{3} - \frac{8 πn}{3}

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( π - 3 θ )}{4} : π - 3 θ

\frac{4 ( π - 3 θ )}{4}

Divide the numbers:

\frac{4}{4} = 1

= π - 3 θ

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 θ = 6 π + 8 πn

π - 3 θ = 6 π + 8 πn

π - 3 θ = 6 π + 8 πn

Move

π

to the right side

π - 3 θ = 6 π + 8 πn

Subtract

π

from both sides

π - 3 θ - π = 6 π + 8 πn - π

Simplify

- 3 θ = 5 π + 8 πn

- 3 θ = 5 π + 8 πn

Divide both sides by

- 3

- 3 θ = 5 π + 8 πn

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 θ}{- 3} : θ

\frac{- 3 θ}{- 3}

Apply the fraction rule:

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

= \frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

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}

θ = - \frac{5 π}{3} - \frac{8 πn}{3}

θ = - \frac{5 π}{3} - \frac{8 πn}{3}

θ = - \frac{5 π}{3} - \frac{8 πn}{3}

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

Combine all the solutions

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

Merge Overlapping Intervals

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

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

cos (\frac{- θ + π}{2}) - cos (θ) = 0

Rewrite using trig identities

cos (\frac{- θ + π}{2}) - cos (θ)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

\frac{\frac{- θ + π}{2} + θ}{2}

Join

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

\frac{- θ + π}{2} + θ

Convert element to fraction:

θ = \frac{θ 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{- θ + π + θ \cdot 2}{2}

- θ + π + θ \cdot 2 = θ + π

- θ + π + θ \cdot 2

Group like terms

= - θ + 2 θ + π

Add similar elements:

- θ + 2 θ = θ

= θ + π

= \frac{θ + π}{2}

= \frac{\frac{θ + π}{2}}{2}

Apply the fraction rule:

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

= \frac{θ + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{θ + π}{4}

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

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

\frac{\frac{- θ + π}{2} - θ}{2}

Join

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

\frac{- θ + π}{2} - θ

Convert element to fraction:

θ = \frac{θ 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{- θ + π - θ \cdot 2}{2}

- θ + π - θ \cdot 2 = - 3 θ + π

- θ + π - θ \cdot 2

Group like terms

= - θ - 2 θ + π

Add similar elements:

- θ - 2 θ = - 3 θ

= - 3 θ + π

= \frac{- 3 θ + π}{2}

= \frac{\frac{- 3 θ + π}{2}}{2}

Apply the fraction rule:

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

= \frac{- 3 θ + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 θ + π}{4}

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

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

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

Solving each part separately

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

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

sin (\frac{θ + π}{4}) = 0

General solutions for

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

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

Solve

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

\frac{θ + π}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{θ + π}{4} = 2 πn

Multiply both sides by

4

\frac{θ + π}{4} = 2 πn

Multiply both sides by

4

\frac{4 ( θ + π )}{4} = 4 \cdot 2 πn

Simplify

θ + π = 8 πn

θ + π = 8 πn

Move

π

to the right side

θ + π = 8 πn

Subtract

π

from both sides

θ + π - π = 8 πn - π

Simplify

θ = 8 πn - π

θ = 8 πn - π

Solve

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

\frac{θ + π}{4} = π + 2 πn

Multiply both sides by

4

\frac{θ + π}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

θ + π = 4 π + 8 πn

θ + π = 4 π + 8 πn

Move

π

to the right side

θ + π = 4 π + 8 πn

Subtract

π

from both sides

θ + π - π = 4 π + 8 πn - π

Simplify

θ = 3 π + 8 πn

θ = 3 π + 8 πn

θ = 8 πn - π, θ = 3 π + 8 πn

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

sin (\frac{π - 3 θ}{4}) = 0

General solutions for

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

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

Solve

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

\frac{π - 3 θ}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{π - 3 θ}{4} = 2 πn

Multiply both sides by

4

\frac{π - 3 θ}{4} = 2 πn

Multiply both sides by

4

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

Simplify

π - 3 θ = 8 πn

π - 3 θ = 8 πn

Move

π

to the right side

π - 3 θ = 8 πn

Subtract

π

from both sides

π - 3 θ - π = 8 πn - π

Simplify

- 3 θ = 8 πn - π

- 3 θ = 8 πn - π

Divide both sides by

- 3

- 3 θ = 8 πn - π

Divide both sides by

- 3

\frac{- 3 θ}{- 3} = \frac{8 πn}{- 3} - \frac{π}{- 3}

Simplify

\frac{- 3 θ}{- 3} = \frac{8 πn}{- 3} - \frac{π}{- 3}

Simplify

\frac{- 3 θ}{- 3} : θ

\frac{- 3 θ}{- 3}

Apply the fraction rule:

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

= \frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

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

\frac{8 πn}{- 3} - \frac{π}{- 3}

Apply the fraction rule:

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

= - \frac{8 πn}{3} - \frac{π}{- 3}

Apply the fraction rule:

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

= - \frac{8 πn}{3} - (- \frac{π}{3})

Apply rule

- (- a) = a

= - \frac{8 πn}{3} + \frac{π}{3}

θ = - \frac{8 πn}{3} + \frac{π}{3}

θ = - \frac{8 πn}{3} + \frac{π}{3}

θ = - \frac{8 πn}{3} + \frac{π}{3}

Solve

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

\frac{π - 3 θ}{4} = π + 2 πn

Multiply both sides by

4

\frac{π - 3 θ}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

π - 3 θ = 4 π + 8 πn

π - 3 θ = 4 π + 8 πn

Move

π

to the right side

π - 3 θ = 4 π + 8 πn

Subtract

π

from both sides

π - 3 θ - π = 4 π + 8 πn - π

Simplify

- 3 θ = 3 π + 8 πn

- 3 θ = 3 π + 8 πn

Divide both sides by

- 3

- 3 θ = 3 π + 8 πn

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 θ}{- 3} : θ

\frac{- 3 θ}{- 3}

Apply the fraction rule:

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

= \frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

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

\frac{3 π}{- 3} + \frac{8 πn}{- 3}

\frac{3 π}{- 3} = - π

\frac{3 π}{- 3}

Apply the fraction rule:

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

= - \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= - π

= - π + \frac{8 πn}{- 3}

Apply the fraction rule:

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

= - π - \frac{8 πn}{3}

θ = - π - \frac{8 πn}{3}

θ = - π - \frac{8 πn}{3}

θ = - π - \frac{8 πn}{3}

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

Combine all the solutions

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

Merge Overlapping Intervals

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

Combine all the solutions

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

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