What is the general solution for sin(pi/6+x)+sin(pi/6-x)= 1/2 ?

The general solution for sin(pi/6+x)+sin(pi/6-x)= 1/2 is x= pi/3+2pin,x=(5pi)/3+2pin

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sin(pi/6+x)+sin(pi/6-x)= 1/2

Solution

sin (\frac{π}{6} + x) + sin (\frac{π}{6} - x) = \frac{1}{2}

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (\frac{π}{6} + x) + sin (\frac{π}{6} - x) = \frac{1}{2}

Rewrite using trig identities

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

Simplify

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

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

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

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

Combine the fractions

\frac{π}{6} + \frac{π}{6} : \frac{π}{3}

Apply rule

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

= \frac{π + π}{6}

Add similar elements:

π + π = 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{\frac{π}{3} + x - x}{2}

Add similar elements:

x - x = 0

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

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

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

Join

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

\frac{π}{6} + x - (\frac{π}{6} - x)

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Expand

π + x \cdot 6 - (\frac{π}{6} - x) \cdot 6 : 12 x

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

= π + 6 x - 6 (\frac{π}{6} - x)

Expand

- 6 (\frac{π}{6} - x) : - π + 6 x

- 6 (\frac{π}{6} - x)

Apply the distributive law:

a (b - c) = ab - a c

a = - 6, b = \frac{π}{6}, c = x

= - 6 \cdot \frac{π}{6} - (- 6) x

Apply minus-plus rules

- (- a) = a

= - 6 \cdot \frac{π}{6} + 6 x

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

= - π + 6 x

= π + x \cdot 6 - π + 6 x

Simplify

π + x \cdot 6 - π + 6 x : 12 x

π + x \cdot 6 - π + 6 x

Group like terms

= 6 x + 6 x + π - π

Add similar elements:

6 x + 6 x = 12 x

= 12 x + π - π

Add similar elements:

π - π = 0

= 12 x

= 12 x

= \frac{12 x}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (\frac{π}{6}) cos (x)

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \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}

= \frac{1}{2}

= 2 \cdot \frac{1}{2} cos (x)

Multiply fractions:

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

= \frac{1 \cdot 2}{2} cos (x)

Cancel the common factor:

2

= cos (x) \cdot 1

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= cos (x)

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{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}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(pi/6+x)+sin(pi/6-x)= 1/2 ?
The general solution for sin(pi/6+x)+sin(pi/6-x)= 1/2 is x= pi/3+2pin,x=(5pi)/3+2pin