What is the general solution for sin(x+pi/6)+sin(x-pi/6)=-(sqrt(3))/2 ?

The general solution for sin(x+pi/6)+sin(x-pi/6)=-(sqrt(3))/2 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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sin(x+pi/6)+sin(x-pi/6)=-(sqrt(3))/2

Solution

sin (x + \frac{π}{6}) + sin (x - \frac{π}{6}) = - \frac{3}{2}

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x + \frac{π}{6}) + sin (x - \frac{π}{6}) = - \frac{3}{2}

Rewrite using trig identities

sin (x + \frac{π}{6}) + sin (x - \frac{π}{6})

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{π}{6} + x - \frac{π}{6}}{2}) cos (\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2})

Simplify

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

2 sin (\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2}) cos (\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2})

\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2} = x

\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2}

x + \frac{π}{6} + x - \frac{π}{6} = 2 x

x + \frac{π}{6} + x - \frac{π}{6}

Group like terms

= x + x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

x + x = 2 x

= 2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{x - ( x - \frac{π}{6} ) + \frac{π}{6}}{2})

\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2} = \frac{π}{6}

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

Join

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

x + \frac{π}{6} - (x - \frac{π}{6})

Convert element to fraction:

x = \frac{x 6}{6}, (x - \frac{π}{6}) = \frac{( x - \frac{π}{6} ) 6}{6}

= \frac{x \cdot 6}{6} + \frac{π}{6} - \frac{( x - \frac{π}{6} ) \cdot 6}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 6 + π - ( x - \frac{π}{6} ) \cdot 6}{6}

Expand

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

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

= 6 x + π - 6 (x - \frac{π}{6})

Expand

- 6 (x - \frac{π}{6}) : - 6 x + π

- 6 (x - \frac{π}{6})

Apply the distributive law:

a (b - c) = ab - a c

a = - 6, b = x, c = \frac{π}{6}

= - 6 x - (- 6) \frac{π}{6}

Apply minus-plus rules

- (- a) = a

= - 6 x + 6 \cdot \frac{π}{6}

6 \cdot \frac{π}{6} = π

6 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 6}{6}

Cancel the common factor:

6

= π

= - 6 x + π

= x \cdot 6 + π - 6 x + π

Simplify

x \cdot 6 + π - 6 x + π : 2 π

x \cdot 6 + π - 6 x + π

Group like terms

= 6 x - 6 x + π + π

Add similar elements:

6 x - 6 x = 0

= π + π

Add similar elements:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

= 2 cos (\frac{π}{6}) sin (x)

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

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}{2}

= 2 \cdot \frac{3}{2} sin (x)

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (x) 3

= 3 sin (x)

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 sin ( x )}{3} : sin (x)

\frac{3 sin ( x )}{3}

Cancel the common factor:

3

= sin (x)

Simplify

\frac{- \frac{3}{2}}{3} : - \frac{1}{2}

\frac{- \frac{3}{2}}{3}

Apply the fraction rule:

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

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

Apply the fraction rule:

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

\frac{\frac{3}{2}}{3} = \frac{3}{2 3}

= - \frac{3}{2 3}

Cancel the common factor:

3

= - \frac{1}{2}

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

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

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

General solutions for

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

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+pi/6)+sin(x-pi/6)=-(sqrt(3))/2 ?
The general solution for sin(x+pi/6)+sin(x-pi/6)=-(sqrt(3))/2 is x=(7pi)/6+2pin,x=(11pi)/6+2pin