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Popular >

sin(2x+60)+sin(x+30)=0

Solution

sin (2 x + 60) + sin (x + 30) = 0

Solution

x = - π - 30 - 4 πn, x = - 3 π - 30 - 4 πn, x = \frac{4 πn}{3} - 30, x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

Degrees

x = - 1898.87338 \dots^{\circ} - 72 0^{\circ} n, x = - 2258.87338 \dots^{\circ} - 72 0^{\circ} n, x = - 1718.87338 \dots^{\circ} + 24 0^{\circ} n, x = - 1598.87338 \dots^{\circ} + 24 0^{\circ} n

Solution steps

sin (2 x + 60) + sin (x + 30) = 0

Rewrite using trig identities

sin (30 + x) + sin (60 + 2 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{30 + x + 60 + 2 x}{2}) cos (\frac{30 + x - ( 60 + 2 x )}{2})

Simplify

2 sin (\frac{30 + x + 60 + 2 x}{2}) cos (\frac{30 + x - ( 60 + 2 x )}{2}) : 2 sin (\frac{3 x + 90}{2}) cos (\frac{- x - 30}{2})

2 sin (\frac{30 + x + 60 + 2 x}{2}) cos (\frac{30 + x - ( 60 + 2 x )}{2})

30 + x + 60 + 2 x = 3 x + 90

30 + x + 60 + 2 x

Group like terms

= x + 2 x + 30 + 60

Add similar elements:

x + 2 x = 3 x

= 3 x + 30 + 60

Add the numbers:

30 + 60 = 90

= 3 x + 90

= 2 sin (\frac{3 x + 90}{2}) cos (\frac{x - ( 2 x + 60 ) + 30}{2})

Expand

30 + x - (60 + 2 x) : - x - 30

30 + x - (60 + 2 x)

- (60 + 2 x) : - 60 - 2 x

- (60 + 2 x)

Distribute parentheses

= - (60) - (2 x)

Apply minus-plus rules

+ (- a) = - a

= - 60 - 2 x

= 30 + x - 60 - 2 x

Simplify

30 + x - 60 - 2 x : - x - 30

30 + x - 60 - 2 x

Group like terms

= x - 2 x + 30 - 60

Add similar elements:

x - 2 x = - x

= - x + 30 - 60

Add/Subtract the numbers:

30 - 60 = - 30

= - x - 30

= - x - 30

= 2 sin (\frac{3 x + 90}{2}) cos (\frac{- x - 30}{2})

= 2 sin (\frac{3 x + 90}{2}) cos (\frac{- x - 30}{2})

2 cos (\frac{- 30 - x}{2}) sin (\frac{90 + 3 x}{2}) = 0

Solving each part separately

cos (\frac{- 30 - x}{2}) = 0 or sin (\frac{90 + 3 x}{2}) = 0

cos (\frac{- 30 - x}{2}) = 0 : x = - π - 30 - 4 πn, x = - 3 π - 30 - 4 πn

cos (\frac{- 30 - x}{2}) = 0

General solutions for

cos (\frac{- 30 - x}{2}) = 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{- 30 - x}{2} = \frac{π}{2} + 2 πn, \frac{- 30 - x}{2} = \frac{3 π}{2} + 2 πn

\frac{- 30 - x}{2} = \frac{π}{2} + 2 πn, \frac{- 30 - x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{- 30 - x}{2} = \frac{π}{2} + 2 πn : x = - π - 30 - 4 πn

\frac{- 30 - x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{- 30 - x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 30 - x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 30 - x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 30 - x )}{2} : - 30 - x

\frac{2 ( - 30 - x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 30 - x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

- 30 - x = π + 4 πn

- 30 - x = π + 4 πn

- 30 - x = π + 4 πn

Move

30

to the right side

- 30 - x = π + 4 πn

Add

30

to both sides

- 30 - x + 30 = π + 4 πn + 30

Simplify

- x = π + 4 πn + 30

- x = π + 4 πn + 30

Divide both sides by

- 1

- x = π + 4 πn + 30

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Simplify

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1} : - π - 30 - 4 πn

\frac{π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Group like terms

= \frac{π}{- 1} + \frac{30}{- 1} + \frac{4 πn}{- 1}

\frac{π}{- 1} = - π

\frac{π}{- 1}

Apply the fraction rule:

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

= - \frac{π}{1}

Apply rule

\frac{a}{1} = a

= - π

= - π + \frac{30}{- 1} + \frac{4 πn}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

Apply the fraction rule:

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

= - \frac{30}{1}

Apply rule

\frac{a}{1} = a

= - 30

= - π - 30 + \frac{4 πn}{- 1}

\frac{4 πn}{- 1} = - 4 πn

\frac{4 πn}{- 1}

Apply the fraction rule:

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

= - \frac{4 πn}{1}

Apply rule

\frac{a}{1} = a

= - 4 πn

= - π - 30 - 4 πn

x = - π - 30 - 4 πn

x = - π - 30 - 4 πn

x = - π - 30 - 4 πn

Solve

\frac{- 30 - x}{2} = \frac{3 π}{2} + 2 πn : x = - 3 π - 30 - 4 πn

\frac{- 30 - x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{- 30 - x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 30 - x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 30 - x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 30 - x )}{2} : - 30 - x

\frac{2 ( - 30 - x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 30 - x

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

- 30 - x = 3 π + 4 πn

- 30 - x = 3 π + 4 πn

- 30 - x = 3 π + 4 πn

Move

30

to the right side

- 30 - x = 3 π + 4 πn

Add

30

to both sides

- 30 - x + 30 = 3 π + 4 πn + 30

Simplify

- x = 3 π + 4 πn + 30

- x = 3 π + 4 πn + 30

Divide both sides by

- 1

- x = 3 π + 4 πn + 30

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{3 π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Simplify

\frac{- x}{- 1} = \frac{3 π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{3 π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1} : - 3 π - 30 - 4 πn

\frac{3 π}{- 1} + \frac{4 πn}{- 1} + \frac{30}{- 1}

Group like terms

= \frac{3 π}{- 1} + \frac{30}{- 1} + \frac{4 πn}{- 1}

\frac{3 π}{- 1} = - 3 π

\frac{3 π}{- 1}

Apply the fraction rule:

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

= - \frac{3 π}{1}

Apply rule

\frac{a}{1} = a

= - 3 π

= - 3 π + \frac{30}{- 1} + \frac{4 πn}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

Apply the fraction rule:

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

= - \frac{30}{1}

Apply rule

\frac{a}{1} = a

= - 30

= - 3 π - 30 + \frac{4 πn}{- 1}

\frac{4 πn}{- 1} = - 4 πn

\frac{4 πn}{- 1}

Apply the fraction rule:

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

= - \frac{4 πn}{1}

Apply rule

\frac{a}{1} = a

= - 4 πn

= - 3 π - 30 - 4 πn

x = - 3 π - 30 - 4 πn

x = - 3 π - 30 - 4 πn

x = - 3 π - 30 - 4 πn

x = - π - 30 - 4 πn, x = - 3 π - 30 - 4 πn

sin (\frac{90 + 3 x}{2}) = 0 : x = \frac{4 πn}{3} - 30, x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

sin (\frac{90 + 3 x}{2}) = 0

General solutions for

sin (\frac{90 + 3 x}{2}) = 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{90 + 3 x}{2} = 0 + 2 πn, \frac{90 + 3 x}{2} = π + 2 πn

\frac{90 + 3 x}{2} = 0 + 2 πn, \frac{90 + 3 x}{2} = π + 2 πn

Solve

\frac{90 + 3 x}{2} = 0 + 2 πn : x = \frac{4 πn}{3} - 30

\frac{90 + 3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{90 + 3 x}{2} = 2 πn

Multiply both sides by

2

\frac{90 + 3 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 90 + 3 x )}{2} = 2 \cdot 2 πn

Simplify

90 + 3 x = 4 πn

90 + 3 x = 4 πn

Move

90

to the right side

90 + 3 x = 4 πn

Subtract

90

from both sides

90 + 3 x - 90 = 4 πn - 90

Simplify

3 x = 4 πn - 90

3 x = 4 πn - 90

Divide both sides by

3

3 x = 4 πn - 90

Divide both sides by

3

\frac{3 x}{3} = \frac{4 πn}{3} - \frac{90}{3}

Simplify

x = \frac{4 πn}{3} - 30

x = \frac{4 πn}{3} - 30

Solve

\frac{90 + 3 x}{2} = π + 2 πn : x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

\frac{90 + 3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{90 + 3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 90 + 3 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

90 + 3 x = 2 π + 4 πn

90 + 3 x = 2 π + 4 πn

Move

90

to the right side

90 + 3 x = 2 π + 4 πn

Subtract

90

from both sides

90 + 3 x - 90 = 2 π + 4 πn - 90

Simplify

3 x = 2 π + 4 πn - 90

3 x = 2 π + 4 πn - 90

Divide both sides by

3

3 x = 2 π + 4 πn - 90

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3} - \frac{90}{3}

Simplify

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3} - \frac{90}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 π}{3} + \frac{4 πn}{3} - \frac{90}{3} : \frac{2 π}{3} - 30 + \frac{4 πn}{3}

\frac{2 π}{3} + \frac{4 πn}{3} - \frac{90}{3}

Group like terms

= \frac{2 π}{3} - \frac{90}{3} + \frac{4 πn}{3}

Divide the numbers:

\frac{90}{3} = 30

= \frac{2 π}{3} - 30 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

x = \frac{4 πn}{3} - 30, x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

Combine all the solutions

x = - π - 30 - 4 πn, x = - 3 π - 30 - 4 πn, x = \frac{4 πn}{3} - 30, x = \frac{2 π}{3} - 30 + \frac{4 πn}{3}

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