What is the general solution for sin((2pi)/3-x)+sin((2pi)/3+x)=0 ?

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

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sin((2pi)/3-x)+sin((2pi)/3+x)=0

Solution

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

Solution

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

Degrees

x = - 9 0^{\circ} - 36 0^{\circ} n, x = - 27 0^{\circ} - 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Simplify

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 π + 2 π}{3}

Add similar elements:

2 π + 2 π = 4 π

= \frac{4 π}{3}

= \frac{\frac{4 π}{3} - x + x}{2}

Add similar elements:

- x + x = 0

= \frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

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

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

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

Join

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

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

Convert element to fraction:

x = \frac{x 3}{3}, (x + \frac{2 π}{3}) = \frac{( \frac{2 π}{3} + x ) 3}{3}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

2 π - x \cdot 3 - (\frac{2 π}{3} + x) \cdot 3 : - 6 x

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

= 2 π - 3 x - 3 (\frac{2 π}{3} + x)

Expand

- 3 (\frac{2 π}{3} + x) : - 2 π - 3 x

- 3 (\frac{2 π}{3} + x)

Apply the distributive law:

a (b + c) = ab + a c

a = - 3, b = \frac{2 π}{3}, c = x

= - 3 \cdot \frac{2 π}{3} + (- 3) x

Apply minus-plus rules

+ (- a) = - a

= - 3 \cdot \frac{2 π}{3} - 3 x

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

= - 2 π - 3 x

= 2 π - x \cdot 3 - 2 π - 3 x

Simplify

2 π - x \cdot 3 - 2 π - 3 x : - 6 x

2 π - x \cdot 3 - 2 π - 3 x

Group like terms

= - 3 x - 3 x + 2 π - 2 π

Add similar elements:

- 3 x - 3 x = - 6 x

= - 6 x + 2 π - 2 π

Add similar elements:

2 π - 2 π = 0

= - 6 x

= - 6 x

= \frac{- 6 x}{3}

Apply the fraction rule:

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

= - \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

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

= - \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= - x

= 2 sin (\frac{2 π}{3}) cos (- x)

Simplify

sin (\frac{2 π}{3}) : \frac{3}{2}

sin (\frac{2 π}{3})

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

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}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (- x) 3

= 3 cos (- x)

3 cos (- x) = 0

Divide both sides by

3

3 cos (- x) = 0

Divide both sides by

3

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

Simplify

cos (- x) = 0

cos (- x) = 0

General solutions for

cos (- x) = 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}

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

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

Solve

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

- x = \frac{π}{2} + 2 πn

Divide both sides by

- 1

- x = \frac{π}{2} + 2 πn

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{π}{2}}{- 1} + \frac{2 πn}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{π}{2}}{- 1} + \frac{2 πn}{- 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{\frac{π}{2}}{- 1} + \frac{2 πn}{- 1} : - \frac{π}{2} - 2 πn

\frac{\frac{π}{2}}{- 1} + \frac{2 πn}{- 1}

\frac{\frac{π}{2}}{- 1} = - \frac{π}{2}

\frac{\frac{π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{π}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{π}{2}}{1} = \frac{π}{2}

= - \frac{π}{2}

= - \frac{π}{2} + \frac{2 πn}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - \frac{π}{2} - 2 πn

x = - \frac{π}{2} - 2 πn

x = - \frac{π}{2} - 2 πn

x = - \frac{π}{2} - 2 πn

Solve

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

- x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 1

- x = \frac{3 π}{2} + 2 πn

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{3 π}{2}}{- 1} + \frac{2 πn}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{3 π}{2}}{- 1} + \frac{2 πn}{- 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{\frac{3 π}{2}}{- 1} + \frac{2 πn}{- 1} : - \frac{3 π}{2} - 2 πn

\frac{\frac{3 π}{2}}{- 1} + \frac{2 πn}{- 1}

\frac{\frac{3 π}{2}}{- 1} = - \frac{3 π}{2}

\frac{\frac{3 π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{3 π}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{3 π}{2}}{1} = \frac{3 π}{2}

= - \frac{3 π}{2}

= - \frac{3 π}{2} + \frac{2 πn}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - \frac{3 π}{2} - 2 πn

x = - \frac{3 π}{2} - 2 πn

x = - \frac{3 π}{2} - 2 πn

x = - \frac{3 π}{2} - 2 πn

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

Popular Examples

sin(θ)cos(θ)=cot(θ)

sin (θ) cos (θ) = cot (θ)

cot(θ)=-4/3

cot (θ) = - \frac{4}{3}

tan(x)= 6/10

tan (x) = \frac{6}{10}

tan(A)=34

tan (A) = 34

sqrt(2)sec(x)=2

2 sec (x) = 2

Frequently Asked Questions (FAQ)

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