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

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

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

Solution

sin (x + \frac{π}{6}) - sin (x - \frac{π}{6}) = \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 (x + \frac{π}{6}) - sin (x - \frac{π}{6}) = \frac{1}{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}) : cos (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} = \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 sin (\frac{π}{6}) cos (\frac{x + x + \frac{π}{6} - \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 (\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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tan (θ) = \frac{5}{12}

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Frequently Asked Questions (FAQ)

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